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2 2 4 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT -1 49 " Transformations of Gauss hypergeometric functions" }}{PARA 3 "" 0 "" {TEXT -1 27 "Copenhagen, August 18, 2003" }}{PARA 3 "" 0 "" {TEXT -1 23 "Kobe, November 22, 2003" }}{PARA 3 "" 0 "" {TEXT -1 23 "Kyto, Dece mber 15, 2003" }}{PARA 4 "" 0 "" {TEXT -1 33 "by Raimundas Vidunas\nVe rsion 3.12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 257 "" 0 " " {TEXT -1 12 "Introduction" }}{PARA 259 "" 0 "" {TEXT -1 85 "An algeb raic transformation of Gauss hypergeometric series is an identity of t he form" }{TEXT 258 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 259 0 "Hypergeom ([alpha, beta],[gamma],x) = theta(x)*Hypergeom([A, B],[C],phi(x));" "6 #/-%*HypergeomG6%7$%&alphaG%%betaG7#%&gammaG%\"xG*&-%&thetaG6#F,\"\"\" -F%6%7$%\"AG%\"BG7#%\"CG-%$phiG6#F,F1" }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 35 "Example (quad ratic transformations)" }{TEXT 260 0 "" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "Hypergeom( [a,b], [(a+b+1)/2], x ) =\nHyp ergeom( [a/2,b/2], [(a+b+1)/2], 4*x*(1-x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$%\"aG%\"bG7#,(*&\"\"#!\"\"F(\"\"\"F/* &F-F.F)F/F/#F/F-F/%\"xG-F%6%7$,$*&F-F.F(F/F/,$*&F-F.F)F/F/F*,$*(\"\"%F /F2F/,&F/F/F2F.F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "Hyp ergeom( [a,(a-b+1)/2], [a-b+1], x ) = (1-x/2)^(-a)*\nHypergeom( [a/2,( a+1)/2], [(a-b)/2+1], x^2/(2-x)^2 );" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#/-%*HypergeomG6%7$%\"aG,(*&\"\"#!\"\"F(\"\"\"F-*&F+F,%\"bGF-F,#F-F+F -7#,(F(F-F/F,F-F-%\"xG*&),&F-F-*&F+F,F3F-F,,$F(F,F--F%6%7$,$*&F+F,F(F- F-,&*&F+F,F(F-F-F0F-7#,(*&F+F,F(F-F-*&F+F,F/F-F,F-F-*&F3F+,&F+F-F3F,! \"#F-" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT 261 36 "Transformations of \+ hypergeometric eq" }{TEXT 450 0 "" }{TEXT 257 7 "uations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 16 "Recall that the " }{TEXT 263 29 "Gauss hypergeometric function" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "series( Hypergeom( [A, B], [C], z ), z=0, 4 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"zG\"\"\"\"\"!*(%\"AGF%%\"BGF%% \"CG!\"\"F%,$*0\"\"#F+F(F%F)F%F*F+,&F(F%F%F%F%,&F)F%F%F%F%,&F*F%F%F%F+ F%F.,$*6\"\"'F+F(F%F)F%F*F+F/F%F0F%F1F+,&F(F%F.F%F%,&F)F%F.F%F%,&F*F%F .F%F+F%\"\"$-%\"OG6#F%\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "sa tisfies the " }{TEXT 265 36 "hypergeometric differential equation" } {TEXT 266 1 ":" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "z* (1-z)*diff(y(z),z$2)+(C-(A+B+1)*z)*diff(y(z),z)-A*B*y(z)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(%\"zG\"\"\",&F'F'F&!\"\"F'-%%diffG6$-% \"yG6#F&-%\"$G6$F&\"\"#F'F'*&,&%\"CGF'*&,(%\"AGF'%\"BGF'F'F'F'F&F'F)F' -F+6$F-F&F'F'*(F9F'F:F'F-F'F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 267 2 "A " }{TEXT 268 24 "pull-back transformation" } {TEXT 269 36 " (with respect to a finite morphism " }{XPPEDIT 18 0 "ph i;" "6#%$phiG" }{TEXT 325 34 ") of such an equation has the form" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "z=phi(x), Y(x)=theta(x)*y(phi(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"zG-%$phiG6#%\"xG/-%\"YGF'*&-%& thetaGF'\"\"\"-%\"yG6#F%F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 106 "If the transformed equation is a hypergeometric equation agai n, we have two-term algebraic transformations" }}{PARA 0 "" 0 "" {TEXT -1 61 "of their hypergeometric equations. For example, if the po int " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 1 " " }{TEXT 272 21 "lies above the point " }{XPPEDIT 18 0 "z = 0;" "6#/%\"zG\"\"! " }{TEXT 273 73 ", then we can identify the hypergeometric series with the local exponent " }{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT -1 97 " at \+ those both points. One can use fractional-linear transformations to fi nd all such situations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 275 238 "Conversely, if we have a two-term hypergeometric identity as abov e, and the right hand side is not a solution of a first order linear h omogeneous differential eqaution, we have a pull-back transformation o f their hypergeometric equations." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 276 195 "In this talk we classify algebraic tran sformations of Gauss hypergeometric functions by considering pull-back transformations of their hypergeometric equations to other hypergeome tric equations. " }{TEXT -1 0 "" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 21 "Classification scheme" }}{PARA 259 "" 0 "" {TEXT -1 53 "A hyper geometric equation is a Fucshian equation on " }{TEXT 316 2 "P1" } {TEXT -1 37 " with three regular singular points " }{XPPEDIT 317 0 "z = 0,1,infinity;" "6%/%\"zG\"\"!\"\"\"%)infinityG" }{TEXT -1 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 318 65 "It is completely characte rized by the local exponent differences " }{XPPEDIT 18 0 "1-C,C-A-B,A- B;" "6%,&\"\"\"F$%\"CG!\"\",(F%F$%\"AGF&%\"BGF&,&F(F$F)F&" }{TEXT 319 17 " at these points." }}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 0 "" } {TEXT 320 60 "The hypergeometric equation with local exponent differen ces " }{XPPEDIT 321 0 "e[0],e[1],e[infinity];" "6%&%\"eG6#\"\"!&F$6#\" \"\"&F$6#%)infinityG" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "dif f(y(z),z$2)+((1-e[0])/z+(1-e[1])/(z-1))*diff(y(z),z)+\n((1-e[0]-e[1])^ 2-e[infinity]^2)/4/z/(z-1)*y(z)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /,(-%%diffG6$-%\"yG6#%\"zG-%\"$G6$F+\"\"#\"\"\"*&,&*&,&F0F0&%\"eG6#\" \"!!\"\"F0F+F9F0*&,&F0F0&F66#F0F9F0,&F+F0F0F9F9F9F0-F&6$F(F+F0F0*&#F0 \"\"%F0**,&*$),(F0F0F5F9FF9F (F0F0F0F8" }}}{PARA 0 "" 0 "" {TEXT 322 41 "Permutation of local expon ent differences" }{TEXT -1 1 " " }{TEXT 256 0 "" }{XPPEDIT -1 0 "e[0], e[1], e[infinity]" "6%&%\"eG6#\"\"!&F$6#\"\"\"&F$6#%)infinityG" } {TEXT -1 1 " " }{TEXT 323 55 "or changing their signs gives a hypergeo metric equation" }}{PARA 0 "" 0 "" {TEXT 324 75 "which is related by a fractional-linear transformation to the original one." }}}{PARA 259 " " 0 "" {TEXT -1 195 "A pull-back transformation of a Fucshian equation is a Fucshian equation again. The singularities and local exponent di fferences of the transformed equation are determined by the finite cov ering " }{XPPEDIT 326 0 "phi;" "6#%$phiG" }{TEXT -1 1 "." }}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 0 "" }{TEXT 327 4 "Let " }{XPPEDIT 399 0 "p hi;" "6#%$phiG" }{TEXT -1 0 "" }{TEXT 331 2 ": " }{TEXT -1 2 "P1" } {XPPEDIT 401 0 "``[x];" "6#&%!G6#%\"xG" }{XPPEDIT 18 0 "-`>`;" "6#,$% \">G!\"\"" }{TEXT -1 3 " P1" }{XPPEDIT 402 0 "``[z];" "6#&%!G6#%\"zG" }{TEXT -1 1 " " }{TEXT 328 32 "denote a finite covering, let " } {XPPEDIT 403 0 "E[1];" "6#&%\"EG6#\"\"\"" }{TEXT 329 32 " denote a Fu chsian equation on " }{TEXT -1 2 "P1" }{XPPEDIT 404 0 "``[z];" "6#&%!G 6#%\"zG" }{TEXT 330 2 " ," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 358 7 "and let" }{TEXT 370 1 " " }{XPPEDIT 18 0 "E[2];" "6#&%\"EG6#\"\"#" }{TEXT -1 1 " " }{TEXT 359 6 "denote" }{TEXT 371 0 "" }{TEXT 357 30 " \+ a pull-back transformation of" }{TEXT 372 1 " " }{XPPEDIT 18 0 "E[1]" "6#&%\"EG6#\"\"\"" }{TEXT 361 1 " " }{TEXT 373 15 "with respect to" } {TEXT 374 1 " " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT 360 2 ".\n" } {TEXT 375 3 "Let" }{TEXT 376 1 " " }{XPPEDIT 18 0 "P;" "6#%\"PG" } {TEXT 364 1 " " }{TEXT 377 3 " in" }{TEXT 378 2 " " }{TEXT 264 2 "P1 " }{TEXT 362 0 "" }{XPPEDIT 18 0 "``[x];" "6#&%!G6#%\"xG" }{TEXT 363 3 " , " }{TEXT 366 0 "" }{TEXT 365 1 " " }{XPPEDIT 18 0 "Q;" "6#%\"QG " }{TEXT 367 2 " " }{TEXT 379 2 "in" }{TEXT 380 2 " " }{TEXT 271 2 " P1" }{XPPEDIT 18 0 "``[z];" "6#&%!G6#%\"zG" }{TEXT 368 2 " " }{TEXT 381 19 "be points such that" }{TEXT 382 2 " " }{XPPEDIT 18 0 "phi(P) \+ = Q;" "6#/-%$phiG6#%\"PG%\"QG" }{TEXT 369 3 ". " }{TEXT 383 4 "Then" }{TEXT 384 1 ":" }{TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 2 "* " } {TEXT 333 13 "If the point " }{XPPEDIT 405 0 "Q;" "6#%\"QG" }{TEXT 332 2 " " }{TEXT 335 23 "is a singular point for" }{TEXT 334 1 " " } {XPPEDIT 406 0 "E[1];" "6#&%\"EG6#\"\"\"" }{TEXT 336 17 ", then the po int " }{XPPEDIT 407 0 "P;" "6#%\"PG" }{TEXT -1 1 " " }{TEXT 337 23 "is a regular point for " }{XPPEDIT 408 0 "E[2];" "6#&%\"EG6#\"\"#" } {TEXT 338 42 " only if\nthe local exponent difference at " }{XPPEDIT 409 0 "Q" "6#%\"QG" }{TEXT -1 1 " " }{TEXT 339 12 "is equal to " } {XPPEDIT 410 0 "1/n;" "6#*&\"\"\"F$%\"nG!\"\"" }{TEXT -1 1 " " }{TEXT 340 8 ", where " }{XPPEDIT 411 0 "n;" "6#%\"nG" }{TEXT -1 1 " " } {TEXT 341 29 "is the ramification index of " }{TEXT -1 1 "\000" } {XPPEDIT 412 0 "phi;" "6#%$phiG" }{TEXT -1 1 " " }{TEXT 342 3 "at " } {XPPEDIT 413 0 "P;" "6#%\"PG" }{TEXT -1 0 "" }{TEXT 343 1 "." }}{PARA 260 "" 0 "" {TEXT -1 2 "* " }{TEXT 345 13 "If the point " }{XPPEDIT 414 0 "Q;" "6#%\"QG" }{TEXT 344 2 " " }{TEXT 347 22 "is a regular poi nt for" }{TEXT 346 1 " " }{XPPEDIT 415 0 "E[1];" "6#&%\"EG6#\"\"\"" } {TEXT 348 16 ", then the point" }{TEXT -1 1 " " }{TEXT 400 0 "" } {XPPEDIT 416 0 "P;" "6#%\"PG" }{TEXT 349 25 " is a regular point for \+ " }{XPPEDIT 417 0 "E[2];" "6#&%\"EG6#\"\"#" }{TEXT 350 22 " only if\nt he covering " }{TEXT 352 1 "\000" }{XPPEDIT 418 0 "phi;" "6#%$phiG" } {TEXT 353 1 " " }{TEXT 356 17 " is unramified at" }{TEXT 355 1 " " } {XPPEDIT 419 0 "P;" "6#%\"PG" }{TEXT 354 0 "" }{TEXT 351 1 "." }} {PARA 260 "" 0 "" {TEXT -1 2 "* " }{TEXT 274 4 "Let " }{XPPEDIT 420 0 "d;" "6#%\"dG" }{TEXT 385 22 " denote the degree of " }{XPPEDIT 421 0 "phi;" "6#%$phiG" }{TEXT 386 10 ", and let " }{XPPEDIT 422 0 "Sigma;" "6#%&SigmaG" }{TEXT 387 33 " denote a set of three points on " }{TEXT 388 2 "P1" }{XPPEDIT 423 0 "``[z];" "6#&%!G6#%\"zG" }{TEXT 389 32 ".\n If all ramification points of " }{XPPEDIT 424 0 "phi;" "6#%$phiG" } {TEXT 390 11 " lie above " }{XPPEDIT 425 0 "Sigma;" "6#%&SigmaG" } {TEXT 391 25 ", then there are exactly " }{XPPEDIT 426 0 "d+2;" "6#,&% \"dG\"\"\"\"\"#F%" }{TEXT 392 20 " distinct points on " }{TEXT 393 2 " P1" }{TEXT 394 0 "" }{XPPEDIT 427 0 "``[x];" "6#&%!G6#%\"xG" }{TEXT 395 7 " above " }{XPPEDIT 428 0 "Sigma;" "6#%&SigmaG" }{TEXT 396 33 ". Otherwise there are more than " }{XPPEDIT 429 0 "d+2" "6#,&%\"dG\"\" \"\"\"#F%" }{TEXT 397 23 " distinct points above " }{XPPEDIT 430 0 "Si gma;" "6#%&SigmaG" }{TEXT 398 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 431 44 "The classification scheme is the following:\n" }{TEXT 434 1 "*" }{TEXT 433 50 " no restriction on the local exponent \+ differences " }{XPPEDIT 18 0 "-` `;" "6#,$%\"~G!\"\"" }{TEXT -1 0 "" } {TEXT 432 38 "only fractional-linear tranformations;" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 435 0 "" }{TEXT -1 0 "" }{TEXT 437 1 "*" } {TEXT 446 47 " only one restricted local exponent difference " }{TEXT 445 0 "" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" }{TEXT -1 0 "" }{TEXT 438 40 "the classical quadratic transformations;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 436 0 "" }{TEXT 439 2 "* " }{TEXT 440 42 "two res tricted local exponent differences " }{XPPEDIT 18 0 "-``;" "6#,$%!G!\" \"" }{TEXT -1 0 "" }{TEXT 441 156 "classical transformations of degree 3, 4, 6, and\n transformations of hypergeometric equations with the multiplicative, cyclic or dihedral monodromy group;" }}{PARA 0 "" 0 " " {TEXT 442 1 "*" }{TEXT 447 54 " all three local exponent differences are restricted, " }{XPPEDIT 18 0 "``(1/k,1/l,1/m);" "6#-%!G6%*&\"\"\" F'%\"kG!\"\"*&F'F'%\"lGF)*&F'F'%\"mGF)" }{TEXT 443 1 " " }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" }{TEXT -1 0 "" }{TEXT 444 38 "transformations \+ of some hypergeometric" }}{PARA 0 "" 0 "" {TEXT 448 106 " equations \+ with a discrete monodromy group; the solutions are algebraic functions , elliptic integrals or" }}{PARA 0 "" 0 "" {TEXT 449 26 " \"hyperbol ic\" functions." }}}}{SECT 0 {PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 551 20 "Algorithmic routines" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 35 " These are algebraic numbers we use:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "alias( omega=RootOf(Z^2+Z+1), beta=RootOf(Z^2+2), xi=RootOf(Z^2+Z+ 2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%&omegaG%%betaG%#xiG" }}} {EXCHG {PARA 259 "" 0 "" {TEXT -1 141 "The following package is for pr esenting expressions of algebraic Gauss hypergoemetric functions. Plea se read it from an appropriate location." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "read \"h21algtr.mpl\";" }}}{PARA 259 "" 0 "" {TEXT -1 33 "The main routines in the package:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# checkseries(Equation, [ Variable[=Expansion Point], NumberOfTerms ] );" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 650 161 "T his routine checks a hypergeometric (or any other kind of) identity by computing the first few terms of the power series for the both sides \+ of a given equality." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# h pgcomposition( FirstEquation, SecondEquation, [ Substitutions ] );" }} {PARA 259 "" 0 "" {TEXT -1 197 "This routine computes a composition of two hypergeometric identities. The parameters of the right-hand side \+ of the first equation should match the ones of the left-hand side of t he second equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "# k ummers24( Hypergeometric2F1Function, FractionalLinearTransformation,\n # [ Variable, [ Substitutions ] ] );" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 651 409 "This routine computes one of the 24 Kummer's soluti ons of the same hypergeometric series. Those solutions are obtained by permuting the three singular points and the local exponents (at the s ame point) of the hypergeometric equation. (There are 48=3!*2*2*2 per mutations and so many hypergeometric functions, if we take into accoun t permutation of the upper parameters, i.e., permuting the local expon ents at " }{XPPEDIT 18 0 "z = infinity;" "6#/%\"zG%)infinityG" }{TEXT 660 16 ". If the point " }{XPPEDIT 18 0 "z = 0;" "6#/%\"zG\"\"!" } {TEXT -1 1 " " }{TEXT 661 310 "and its local exponents are fixed, then we have a two-term fractional-linear transformations.)\nThe second ar gument represents a permutation of this kind. It must be is a permutat ion of the character string \"abc\", and some of these characters may \+ be capital. The character 'a' or 'A' represents the \"new\" point " } {XPPEDIT 18 0 "z = 0;" "6#/%\"zG\"\"!" }{TEXT 653 28 ", the character 'b' or 'B' " }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" }{TEXT 655 16 "the \+ \"new\" point " }{XPPEDIT 18 0 "z = 1;" "6#/%\"zG\"\"\"" }{TEXT 656 161 ", etc. A capital letter represents permutation of the local expon ents at the corresponding point. The first character represents (the f iber of) the \"old\" point " }{XPPEDIT 18 0 "z = 0;" "6#/%\"zG\"\"!" }{TEXT 658 23 ", the middle character " }{XPPEDIT 18 0 "-``;" "6#,$%!G !\"\"" }{TEXT 657 16 "the \"old\" point " }{XPPEDIT 18 0 "z = 1;" "6#/ %\"zG\"\"\"" }{TEXT 659 149 ", etc.\nIf the routine is called with two arguments, the Kummer's solutions are produced up to a scalar multipl e!\nIf it is called with a variabe name " }{TEXT -1 0 "" }{TEXT 654 0 "" }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT 652 57 " as the third argument , the local exponents at the point " }{XPPEDIT 18 0 "z = 0;" "6#/%\"zG \"\"!" }{TEXT -1 1 " " }{TEXT 662 58 "are ignored, and the output is a function which has value " }{XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT -1 1 " " }{TEXT 664 3 "at " }{XPPEDIT 18 0 "z = 0;" "6#/%\"zG\"\"!" } {TEXT -1 0 "" }{TEXT 663 141 ".\nThe remaining extra parameters (if pr esent) are interpreted as substitutions to be perfomed in radical fact ors to hypergeometric functions." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 61 "# alterhpgid( Hypergeometric2F1Identity, [ Sub stitutions ] );" }}{PARA 259 "" 0 "" {TEXT 665 61 "Whenever we have a \+ two-term valid hypergeometric identity of " }{XPPEDIT 670 0 "``[2]*F[1 ];" "6#*&&%!G6#\"\"#\"\"\"&%\"FG6#F(F(" }{TEXT 669 105 ", we can get a new valid identity by considering the second hypergeometric series so lutions at the point " }{XPPEDIT 671 0 "x = 0;" "6#/%\"xG\"\"!" } {TEXT -1 1 " " }{TEXT 666 5 "(or " }{XPPEDIT 672 0 "z = 0;" "6#/%\"zG \"\"!" }{TEXT 667 44 ", etc.) of the both sides. See Lemma 4.1 in " } {TEXT 668 86 "R.V. \"Transformations of some Gauss hypergeometric func tions\", arXiv:math.CA/0310436.\n" }{TEXT 673 43 "This routine compute s that \"dual\" identity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "# otherhpgid( Hypergeometric2F1Identity, LHSTransformation,\n# \+ RHSTransformation, [ Substitutions ] );" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 674 136 "This routine transforms both sides of a hyperg eometric identity by fractional-linear transformations in the most gen eral way (the point " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 0 "" }{TEXT 675 245 " may be moved; correctness of the new identity is not guaranteed, but all two-term hypergeometric identites between \+ the 24 solutions van be obtained from one identity by this routine).\n The two transformations are given by \"abc\"-strings as above." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "# presenthyperid( SchwartzT ypeIdentificationNumber );\n# presenthyperids( SchwartzTypeIdentificat ionNumber );\n# presenthyperidz( SchwartzTypeIdentificationNumber );\n # subsdarboux;" }}{PARA 259 "" 0 "" {TEXT -1 726 "These routines repre sent the explicit Darboux evaluations of the simplest algebraic hyperg eometric solutions of the standard hypergeometric equations with tetra hedral, octahedral or icosahedral monodromy groups. For each Schwartz \+ type there are two identification numbers (the odd one is for the most simplest hypergeometric evaluation, and the even one is for the \"dua l\" evaluation). Identification numbers 1-4 represent tetrahedral type s, numbers 5-8 represent octahedraltypes, numbers 9-28 represent icosa hedral types. The first routine returns only one Darboux evaluation, t he other two - also a contiguous evaluation (in two different forms). \nThe global variable \"subsdarboux\" is a substitution for the Darbou x morphism. " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 19 "The Transformations" }{TEXT 277 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 35 "Classical algeb raic transformations" }}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 33 "Fractio nal-linear transformations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "H21:= Hypergeom( [a,b], [c], x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$H21G-%*HypergeomG6%7$%\"aG%\"bG7#%\"cG%\"xG" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 27 "Pfaff-Kummer transformation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "PfKu:= H21=kummers24(H21,\"aCB\",x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PfKuG/-%*HypergeomG6%7$%\"aG%\"bG7#%\"cG% \"xG*&),&\"\"\"F2F.!\"\",$F*F3F2-F'6%7$F*,&F-F2F+F3F,*&F.F2,&F.F2F2F3F 3F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 451 22 "Euler's tr ansformation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Euler:= H21=kummers 24(H21,\"aBC\",x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&EulerG/-%*Hyp ergeomG6%7$%\"aG%\"bG7#%\"cG%\"xG*&),&\"\"\"F2F.!\"\",(F-F2F*F3F+F3F2- F'6%7$,&F-F2F*F3,&F-F2F+F3F,F.F2" }}}{PARA 0 "" 0 "" {TEXT 644 162 "We can get all fractional-linear transformations by keeping the same fi rst character in the \"abc\"-string and permuting and/or capitilizing \+ the other two letters." }{TEXT -1 0 "" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 147 "Here we illustrate computation of the \"dual\" identity. First we compute as the identity for the second solutions of the hype rgeometric equation at " }{XPPEDIT 676 0 "x = 0;" "6#/%\"xG\"\"!" } {TEXT -1 23 ", with local exponents." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "kummers24( H21, \"Abc\") = kummers24( H21, \"ABC\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&)%\"xG,&\"\"\"F(%\"cG!\"\"F(-%*HypergeomG 6%7$,(%\"aGF(F(F(F)F*,(%\"bGF(F(F(F)F*7#,&\"\"#F(F)F*F&F(*(F%F(),&F(F( F&F*,(F)F(F0F*F2F*F(-F,6%7$,&F(F(F0F*,&F(F(F2F*F3F&F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 645 81 "Here is how we can get th is identity (with ignored local exponent differences at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 0 "" }{TEXT 646 11 ") directly ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "otherhpgid( H21=H21, \"Abc\", \+ \"ABC\" );\nalterhpgid( Euler );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/- %*HypergeomG6%7$,(%\"aG\"\"\"F*F*%\"cG!\"\",(%\"bGF*F*F*F+F,7#,&\"\"#F *F+F,%\"xG*&),&F*F*F2F,,(F+F*F)F,F.F,F*-F%6%7$,&F*F*F)F,,&F*F*F.F,F/F2 F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$,(%\"aG\"\"\"F *F*%\"cG!\"\",(%\"bGF*F*F*F+F,7#,&\"\"#F*F+F,%\"xG*&),&F*F*F2F,,(F+F*F )F,F.F,F*-F%6%7$,&F*F*F)F,,&F*F*F.F,F/F2F*" }}}{PARA 259 "" 0 "" {TEXT -1 300 "In this worksheet, we present all hypergeometric transfo rmations (except those infinitely many identities with degenerate, dih edral, algebraic functions and of hypergeometric integrals) up to two- term fractional-linear transformations, and up to the switch to \"dual \" identity demonstrated just above." }}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 26 "Quadratic transformations " }{XPPEDIT 18 0 "` `(1/2,p,q); " "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"%\"pG%\"qG" }{TEXT -1 3 " <" } {XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` \+ `(p,p,2*q);" "6#-%\"~G6%%\"pGF&*&\"\"#\"\"\"%\"qGF)" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 20 "We already saw them." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Quadr1:= Hypergeom( [a,b], [(a+b+1)/2], x ) =\nHyperg eom( [a/2,b/2], [(a+b+1)/2], 4*x*(1-x) );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Quadr1G/-%*HypergeomG6%7$%\"aG%\"bG7#,(*&\"\"#!\"\"F *\"\"\"F1*&F/F0F+F1F1#F1F/F1%\"xG-F'6%7$,$*&F/F0F*F1F1,$*&F/F0F+F1F1F, ,$*(\"\"%F1F4F1,&F1F1F4F0F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 647 70 "The second familiar identity can be obtained by a 'ot herhpgid' trick: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Quadr2:= other hpgid( Quadr1, \"CbA\", \"CaB\" );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%'Quadr2G/-%*HypergeomG6%7$%\"aG,(*&\"\"#!\"\"F*\"\"\"F/*&F-F.%\"bGF/ F.#F/F-F/7#,(F/F/F1F.F*F/%\"xG*&),&*&F-F.F5F/F.F/F/,$F*F.F/-F'6%7$,$*& F-F.F*F/F/,&F2F/*&F-F.F*F/F/7#,(F/F/*&F-F.F1F/F.*&F-F.F*F/F/*&F5F-,&F5 F/F-F.!\"#F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 648 116 "The following ' otherhpgid' application is a legal swith to (fractional-linear transfo ration) of the \"dual\" identity." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "otherhpgid( Quadr2, \"Abc\", \"ACb\" );\ncheckseries( %, x, 4 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$%\"bG,(*&\"\"#!\" \"%\"aG\"\"\"F,*&F+F,F(F.F.#F.F+F.7#,(F(F.F.F.F-F,%\"xG*&),&F.F.F3F,,$ *&F+F,F(F.F,F.-F%6%7$,&F0F.*&F+F,F-F.F,,$*&F+F,F(F.F.7#,(*&F+F,F(F.F.F .F.*&F+F,F-F.F,,$*(\"\"%F,F3F+,&F3F.F.F,F,F.F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+-%\"xG\"\"\"\"\"!,$*&\"\"#!\"\"%\"bGF&F&F&,$*,\"\")F+ ,(F,F&\"\"$F&%\"aGF+F&,&F,F&F&F&F&F,F&,(F,F&F*F&F2F+F+F&F*,$*.\"#[F+F, F&F3F&,&F,F&F*F&F&,(F2F+F,F&\"\"&F&F&F4F+F&F1-%\"OG6#F&\"\"%F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 0 "" }{TEXT 278 0 "" }{TEXT -1 25 "Degree 3 transformation s " }{XPPEDIT 18 0 "` `(1/2,1/3,p);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*& F'F'\"\"$F)%\"pG" }{TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/2,p,2*p);" "6#-%\"~G6%*&\"\" \"F'\"\"#!\"\"%\"pG*&F(F'F*F'" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "Cubic1:= Hypergeom([a,a+1/2],[(4*a+5)/6],x)=(1+3*x)^(-a)*\nHype rgeom([a/3,(a+1)/3],[(4*a+5)/6],27*x*(x-1)^2/(3*x+1)^3);\ncheckseries( %, x, 4 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Cubic1G/-%*Hypergeom G6%7$%\"aG,&F*\"\"\"#F,\"\"#F,7#,&*(F.F,\"\"$!\"\"F*F,F,#\"\"&\"\"'F,% \"xG*&),&F,F,*&F2F,F7F,F,,$F*F3F,-F'6%7$,$*&F2F3F*F,F,,&#F,F2F,*&F2F3F *F,F,F/,$**\"#FF,F7F,,&F7F,F,F3F.F:!\"$F,F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+-%\"xG\"\"\"\"\"!,$**\"\"$F&%\"aGF&,&*&\"\"#F&F+F&F&F &F&F&,&*&\"\"%F&F+F&F&\"\"&F&!\"\"F&F&,$*2\"\"*F&F.F3F+F&F,F&,&F&F&F+F &F&,&*&F.F&F+F&F&F*F&F&F/F3,&*&F1F&F+F&F&\"#6F&F3F&F.,$*8F6F&F.F3F+F&F ,F&F7F&F8F&,&F+F&F.F&F&,&*&F.F&F+F&F&F2F&F&F/F3F:F3,&*&F1F&F+F&F&\"# " 0 "" {MPLTEXT 1 0 68 "Cubic2:= otherhpgid( Cubic1, \"bac\", \"abc\" );\nche ckseries( %, x, 4 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Cubic2G/-%* HypergeomG6%7$%\"aG,&F*\"\"\"#F,\"\"#F,7#,&*(\"\"%F,\"\"$!\"\"F*F,F,#F .F3F,%\"xG*&),&F,F,*(F3F,F2F4F6F,F4,$F*F4F,-F'6%7$,$*&F3F4F*F,F,,&#F,F 3F,*&F3F4F*F,F,7#,&*(F.F,F3F4F*F,F,#\"\"&\"\"'F,,$**\"#FF,,&F6F,F,F4F, F6F.,&F2F4*&F3F,F6F,F,!\"$F,F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+-% \"xG\"\"\"\"\"!,$*(\"\"$F&\"\"%!\"\"%\"aGF&F&F&,$*.\"\"*F&\"#;F,,&*&\" \"#F&F-F&F&F*F&F&F-F&,&F&F&F-F&F&,&*&F+F&F-F&F&\"\"&F&F,F&F4,$*0F0F&\" $G\"F,F5F&F-F&F2F&,&*&F4F&F-F&F&F8F&F&F6F,F&F*-%\"OG6#F&F+F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Cubic3:= otherhpgid( Cubic1, \"Cba\", \"bac\" );\ncheckseries( %, x, 4 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Cubic3G/-%*HypergeomG6%7$,&#\"\"\"\"\"'F,*&\"\"$!\" \"%\"aGF,F,F17##F,\"\"#%\"xG*&),&*&F/F0F5F,F,F,F,,$F1F0F,-F'6%7$,$*&F/ F0F1F,F,,&#F,F/F,*&F/F0F1F,F,F2*(,&\"\"*F0F5F,F4F5F,,&F5F,F/F,!\"$F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+-%\"xG\"\"\"\"\"!,$*(\"\"$!\"\"%\" aGF&,&*&\"\"#F&F,F&F&F&F&F&F&F&,$*,\"#aF+F-F&F,F&,&\"\"(F&*&F/F&F,F&F& F&,&F&F&F,F&F&F&F/,$*0\"%ICF+F-F&F,F&F3F&F6F&,&\"#8F&*&F/F&F,F&F&F&,&F ,F&F/F&F&F&F*-%\"OG6#F&\"\"%F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 25 "Degree 3 transform ations " }{XPPEDIT 18 0 "` `(1/3,1/3,p);" "6#-%\"~G6%*&\"\"\"F'\"\"$! \"\"*&F'F'F(F)%\"pG" }{TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%!G! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(p,p,p);" "6#-%\"~G6%%\"pGF&F &" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 12 "Recall that " }{XPPEDIT 677 0 "omega;" "6#%&omegaG" }{TEXT -1 14 " is a root of " }{XPPEDIT 678 0 "omega^2+omega+1 = 0;" "6#/,(*$%&omegaG\"\"#\"\"\"F&F(F(F(\"\"! " }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "Cubic4:= Hyper geom([a,(a+1)/3],[(2*a+2)/3],x)=(1+omega^2*x)^(-a)*Hypergeom(\n[a/3,(a +1)/3],[(2*a+2)/3],3*(2*omega+1)*x*(x-1)/(x+omega)^3);\ncheckseries( % , x, 4 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Cubic4G/-%*HypergeomG6 %7$%\"aG,&#\"\"\"\"\"$F-*&F.!\"\"F*F-F-7#,&*(\"\"#F-F.F0F*F-F-#F4F.F-% \"xG*&),&F-F-*&)%&omegaGF4F-F6F-F-,$F*F0F--F'6%7$,$*&F.F0F*F-F-F+F1,$* ,F.F-,&*&F4F-F " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 25 "Degree 4 transformations " }{XPPEDIT 18 0 "` `(1/2,1 /3,p);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)%\"pG" }{TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "` `(1/3,p,3*p);" "6#-%\"~G6%*&\"\"\"F'\"\"$!\"\"%\"pG*& F(F'F*F'" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "Quartic1:= Hype rgeom([4*a/3,(4*a+1)/3],[(4*a+5)/6],x)=(1+8*x)^(-a)*\nHypergeom([a/3,( a+1)/3],[(4*a+5)/6],64*x*(1-x)^3/(8*x+1)^3);\ncheckseries( %, x, 4 ); \+ " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)Quartic1G/-%*HypergeomG6%7$,$*( \"\"%\"\"\"\"\"$!\"\"%\"aGF-F-,&*(F,F-F.F/F0F-F-#F-F.F-7#,&*(\"\"#F-F. F/F0F-F-#\"\"&\"\"'F-%\"xG*&),&F-F-*&\"\")F-F;F-F-,$F0F/F--F'6%7$,$*&F .F/F0F-F-,&F3F-*&F.F/F0F-F-F4,$**\"#kF-F;F-,&F-F-F;F/F.F>!\"$F-F-" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/+-%\"xG\"\"\"\"\"!,$*,\"\")F&\"\"$!\" \"%\"aGF&,&*&\"\"%F&F-F&F&F&F&F&,&*&F0F&F-F&F&\"\"&F&F,F&F&,$*2\"#KF& \"\"*F,F-F&F.F&,&*&F0F&F-F&F&F+F&F&,&F&F&F-F&F&F1F,,&*&F0F&F-F&F&\"#6F &F,F&\"\"#,$*8\"$G\"F&\"#\")F,F-F&F.F&F8F&F:F&,&*&F>F&F-F&F&F+F&F&,&*& F0F&F-F&F&\"\"(F&F&F1F,F;F,,&*&F0F&F-F&F&\"# " 0 "" {MPLTEXT 1 0 72 "Quartic2:= otherhpgid( Qua rtic1, \"bac\", \"abc\" );\ncheckseries( %, x, 4 );" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%)Quartic2G/-%*HypergeomG6%7$,$*(\"\"%\"\"\"\"\"$!\" \"%\"aGF-F-,&*(F,F-F.F/F0F-F-#F-F.F-7#,&*&\"\"#F-F0F-F-#F-F7F-%\"xG*&) ,&F-F-*(\"\")F-\"\"*F/F9F-F/,$F0F/F--F'6%7$,$*&F.F/F0F-F-,&F3F-*&F.F/F 0F-F-7#,&*(F7F-F.F/F0F-F-#\"\"&\"\"'F-,$**\"#kF-,&F9F-F-F/F-F9F.,&F?F/ *&F>F-F9F-F-!\"$F-F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+-%\"xG\"\"\" \"\"!,$*(\"\")F&\"\"*!\"\"%\"aGF&F&F&,$**\"#KF&\"#\")F,F-F&,&F&F&F-F&F &F&\"\"#,$*0\"$G\"F&\"%(=#F,F-F&F2F&,&*&F3F&F-F&F&\"\"$F&F&,&*&\"\"%F& F-F&F&\"\"(F&F&,&*&F=F&F-F&F&\"\"&F&F,F&F:-%\"OG6#F&F=F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Quartic3:= otherhpgid( Quartic1, \" Cba\", \"Cba\" );\ncheckseries( %, x, 4 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)Quartic3G/-%*HypergeomG6%7$,&#\"\"\"\"\"'F,*(\"\"#F, \"\"$!\"\"%\"aGF,F,,$*(\"\"%F,F0F1F2F,F,7##F/F0%\"xG*&),&F,F,F8F1,$F2F 1F,-F'6%7$,&F+F,*&F0F1F2F,F1,$*&F0F1F2F,F,F6,$**\"#kF1,&F8F,\"\")F,F0F 8F,,&F8F,F,F1!\"$F,F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+-%\"xG\"\" \"\"\"!,$*(\"\"$!\"\"%\"aGF&,&*&\"\"%F&F,F&F&F&F&F&F&F&,$*,\"$!=F+F-F& F,F&,&*&F/F&F,F&F&\"\"(F&F&,&*&F/F&F,F&F&F*F&F&F&\"\"#,$*0\"&gH\"F+F-F &F,F&F3F&F6F&,&\"#8F&*&F/F&F,F&F&F&,&*&F8F&F,F&F&F*F&F&F&F*-%\"OG6#F&F /F$" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 25 "Degree 4 transformatio ns " }{XPPEDIT 18 0 "` `(1/2,1/4,p);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"* &F'F'\"\"%F)%\"pG" }{TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(p,p,2*p);" "6#-%\"~G6%%\"pGF&* &\"\"#\"\"\"F&F)" }}{PARA 0 "" 0 "" {TEXT 679 54 "This is a compositio n fo two quadratic transformations" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "Quartic4:= hpgcomposition( subs(b=1/2,Quadr2 ),\nsubs(\{a=a/2,b=a/2+1/2\},Quadr1) ); \ncheckseries(%,x,4);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%)Quartic4G/-%*HypergeomG6%7$%\"aG,&* &\"\"#!\"\"F*\"\"\"F/#F/\"\"%F/7#,&F*F/#F/F-F/%\"xG*&),&*&F-F.F5F/F.F/ F/,$F*F.F/-F'6%7$,$*&F1F.F*F/F/,&F0F/*&F1F.F*F/F/7#,&#\"\"$F1F/*&F-F.F *F/F/,$**\"#;F/,&F5F/F/F.F/F5F-,&F5F/F-F.!\"%F.F/" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+-%\"xG\"\"\"\"\"!,$*&\"\"#!\"\"%\"aGF&F&F&,$*,\"\")F+ ,&*&F*F&F,F&F&\"\"&F&F&F,F&,&F&F&F,F&F&,&*&F*F&F,F&F&\"\"$F&F+F&F*,$*. \"#[F+F3F&F,F&,&F,F&F*F&F&,&*&F*F&F,F&F&\"\"*F&F&F4F+F&F6-%\"OG6#F&\" \"%F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Quartic5:= otherhp gid( Quartic4, \"Cba\", \"abc\" );\ncheckseries(%,x,4);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%)Quartic5G/-%*HypergeomG6%7$#\"\"\"\"\"#%\"aG7 #,&#\"\"$\"\"%F+*&F,!\"\"F-F+F+%\"xG*&),&F+F+*&F,F+F5F+F4,$F-F4F+-F'6% 7$,$*&F2F4F-F+F+,&#F+F2F+*&F2F4F-F+F+F.,$**\"#;F+,&F5F+F+F4F+F5F+,&*&F ,F+F5F+F+F+F4!\"%F+F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+-%\"xG\"\" \"\"\"!,$*(\"\"#F&%\"aGF&,&*&F*F&F+F&F&\"\"$F&!\"\"F&F&,$*,\"\"'F&F+F& ,&F&F&F+F&F&F,F/,&\"\"(F&*&F*F&F+F&F&F/F&F*,$*0\"#?F&F+F&F3F&,&F+F&F*F &F&F,F/F4F/,&\"#6F&*&F*F&F+F&F&F/F&F.-%\"OG6#F&\"\"%F$" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 25 "Degree 6 transformations " }{XPPEDIT 18 0 "` `(1/2,1/3,p);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)%\" pG" }{TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(p,p,4*p);" "6#-%\"~G6%%\"pGF&*&\"\"%\"\"\"F& F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 452 55 "This is a compositi on of degree 2 and 3 transformations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Sextic1:= hpgcomposition( subs(a=2*a,b=2/3*(1-a),Quad r2), Cubic2 ); \ncheckseries(%,x,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#>%(Sextic1G/-%*HypergeomG6%7$,$*&\"\"#\"\"\"%\"aGF-F-,&*(\"\"%F-\"\" $!\"\"F.F-F-#F-\"\"'F-7#,&#F-F2F-*(\"\")F-F2F3F.F-F-%\"xG*&),(*&#F-\"# ;F-*$)F;F,F-F-F-F;F3F-F-,$F.F3F--F'6%7$,$*&F2F3F.F-F-,&F8F-*&F2F3F.F-F -7#,&*(F,F-F2F3F.F-F-#\"\"&F5F-,$**\"$3\"F-,&F;F-F-F3F-F;F1,(FBF-*&FAF -F;F-F3FAF-!\"$F3F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+-%\"xG\"\"\" \"\"!%\"aGF&,$*(\"#;!\"\"F(F&,&*&\"\")F&F(F&F&\"\"(F&F&F&\"\"#,$**\"#[ F,,&*&F/F&F(F&F&\"#8F&F&F(F&,&F&F&F(F&F&F&\"\"$-%\"OG6#F&\"\"%F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sextic2:= otherhpgid( Sextic 1, \"Cba\", \"abc\" );\ncheckseries(%,x,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Sextic2G/-%*HypergeomG6%7$,&#\"\"#\"\"$\"\"\"*(F,F.F -!\"\"%\"aGF.F0,$*&F,F.F1F.F.7#,&*(F,F.F-F0F1F.F.#\"\"&\"\"'F.%\"xG*&) ,(F.F.*&\"#;F.F:F.F0*&F?F.)F:F,F.F.,$F1F0F.-F'6%7$,$*&F-F0F1F.F.,&#F.F -F.*&F-F0F1F.F.F4,$**\"$3\"F.,&F:F.F.F0F.F:F.F=!\"$F.F." }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/+-%\"xG\"\"\"\"\"!,$**\"\")F&,&F&!\"\"%\"aGF&F& F-F&,&*&\"\"%F&F-F&F&\"\"&F&F,F,F&,$*0F*F&F+F&F-F&,&F1F,*&\"\"#F&F-F&F &F&,&*&F6F&F-F&F&F&F&F&F.F,,&*&F0F&F-F&F&\"#6F&F,F&F6,$*8\"#kF&\"\"$F, F+F&F-F&F4F&F7F&,&F0F,F-F&F&,&F&F&F-F&F&F.F,F9F,,&*&F0F&F-F&F&\"# " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 260 "" 0 "" {TEXT -1 25 "Degree 6 transformations " } {XPPEDIT 18 0 "` `(1/2,1/3,p);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F' \"\"$F)%\"pG" }{TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "` `(2*p,2*p,2*p);" "6#-%\"~G6%*&\"\"#\" \"\"%\"pGF(*&F'F(F)F(*&F'F(F)F(" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 453 130 "This is a composition of degree 2 and 3 \+ transformations\n(in two different ways, with essentially different cu bic transformations!)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "Sex tic3:= hpgcomposition( Quadr2, subs(a=a/2,Cubic1), b=(a+1)/3 ); \nchec kseries(%,x,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Sextic3G/-%*Hype rgeomG6%7$%\"aG,&#\"\"\"\"\"$F-*&F.!\"\"F*F-F-7#,&*(\"\"#F-F.F0F*F-F-# F4F.F-%\"xG*&),(*$)F6F4F-F-F6F0F-F-,$*&F4F0F*F-F0F--F'6%7$,$*&\"\"'F0F *F-F-,&F,F-*&FCF0F*F-F-7#,&*&F.F0F*F-F-#\"\"&FCF-,$*,\"#FF-\"\"%F0,&F6 F-F-F0F4F6F4F9!\"$F-F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+-%\"xG\"\" \"\"\"!,$*&\"\"#!\"\"%\"aGF&F&F&,$*,\"\"%F+,&F/F&F,F&F&F,F&,&F&F&F,F&F &,&*&F*F&F,F&F&\"\"&F&F+F&F*,$*.\"#CF+F1F&F,F&,&F,F&F*F&F&,&\"\"(F&F,F &F&F2F+F&\"\"$-%\"OG6#F&F/F$" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Sextic4:= hpgcomposition( Cubic 4, subs(a=a/3,b=(1-a)/3,Quadr2) );" }}{PARA 0 "" 0 "" {TEXT 680 30 "Th is is the other composition." }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Sextic4G/-%*HypergeomG6%7$%\"aG,&#\"\"\"\"\"$F-*&F.! \"\"F*F-F-7#,&*(\"\"#F-F.F0F*F-F-#F4F.F-%\"xG**),&*&F4F0F6F-F0F-F-,$*& F.F0F*F-F0F-),&F-F-*&F4F-F6F-F0F;F-),&F6F-F-F-F;F--F'6%7$,$*&\"\"'F0F* F-F-,&#F-F4F-*&FGF0F*F-F-7#,&*&F.F0F*F-F-#\"\"&FGF-,$*.\"#FF-,&F6F-F-F 0F4F6F4,&F6F-F4F0!\"#FAFU,&*&F4F-F6F-F-F-F0FUF0F-" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 681 74 "Apply a fractional-linear transfor mation and compare the two compositions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "otherhpgid( Sextic4, \"abc\", \"aCB\" );\nevalb(%=Sex tic3);" }}{PARA 259 "" 0 "" {TEXT -1 97 "So the compositions 'Sextic3' and 'Sextic4' are the same up to fractional-linear transformations." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$%\"aG,&#\"\"\"\"\" $F+*&F,!\"\"F(F+F+7#,&*(\"\"#F+F,F.F(F+F+#F2F,F+%\"xG*&),(*$)F4F2F+F+F 4F.F+F+,$*&F2F.F(F+F.F+-F%6%7$,$*&\"\"'F.F(F+F+,&F*F+*&FAF.F(F+F+7#,&* &F,F.F(F+F+#\"\"&FAF+,$*,\"#FF+\"\"%F.,&F4F+F+F.F2F4F2F7!\"$F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 35 "Degenera te hypergeometric functions" }}{PARA 259 "" 0 "" {TEXT -1 52 "Suppose \+ that one local exponent difference (say, at " }{XPPEDIT 465 0 "z = 0; " "6#/%\"zG\"\"!" }{TEXT -1 86 ") is equal to 1.\nTransformations to p ossibly hypergeometric equations have the form " }{XPPEDIT 454 0 "1- z-`>`;" "6#,(\"\"\"F$%\"zG!\"\"%\">GF&" }{TEXT 455 1 " " }{XPPEDIT 456 0 "(1-x)^n;" "6#),&\"\"\"F%%\"xG!\"\"%\"nG" }{TEXT -1 275 " (up to fractional-linear transformations). Additional analysis shows that th e transformed equation is indeed a hypergeometric equation if and only if the point with local exponent difference is a regular point. Then \+ the other two ocal exponenet differences have to be equal." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "Hypergeom( [1+a,1],[2],z)=(1-(1-z)^ (-a))/a/z, a<>0;\nHypergeom( [1,1],[2],z)=log(1-z)/z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%*HypergeomG6%7$,&%\"aG\"\"\"F*F*F*7#\"\"#%\"zG* (,&F*F*),&F*F*F-!\"\",$F)F2F2F*F)F2F-F20F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$\"\"\"F(7#\"\"#%\"zG*&-%#lnG6#,&F(F(F +!\"\"F(F+F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 457 30 "Transfor mations have the form " }{XPPEDIT 18 0 "``(1-z)-`>`;" "6#,&-%!G6#,&\" \"\"F(%\"zG!\"\"F(%\">GF*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-x)^n;" " 6#),&\"\"\"F%%\"xG!\"\"%\"nG" }{TEXT -1 1 " " }{TEXT 458 5 ", or " } {XPPEDIT 18 0 "z-`>`;" "6#,&%\"zG\"\"\"%\">G!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "1-(1-x)^n;" "6#,&\"\"\"F$),&F$F$%\"xG!\"\"%\"nGF(" } {TEXT -1 0 "" }{TEXT 459 1 " " }{XPPEDIT 18 0 "`` = x*phi(x);" "6#/%!G *&%\"xG\"\"\"-%$phiG6#F&F'" }{TEXT 460 1 "." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 81 "Eq:= Hypergeom( [1+n*a,1],[2],x)=phi(x)/n*Hypergeom ( [1+a,1],[2],z):\nEq, `any`*a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-% *HypergeomG6%7$,&\"\"\"F)*&%\"nGF)%\"aGF)F)F)7#\"\"#%\"xG*(-%$phiG6#F/ F)F+!\"\"-F%6%7$,&F,F)F)F)F)F-%\"zGF)*&%$anyGF)F,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "checkseries( subs(phi(x)=z/x,z=1-(1-x)^n, Eq), x, 4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+-%\"xG\"\"\"\"\"!,&#F &\"\"#F&*(F*!\"\"%\"nGF&%\"aGF&F&F&,$*(\"\"'F,,&F&F&*&F-F&F.F&F&F&,&F* F&F3F&F&F&F*,$**\"#CF,F2F&F4F&,&\"\"$F&F3F&F&F&F9-%\"OG6#F&\"\"%F$" }} }{SECT 1 {PARA 260 "" 0 "" {TEXT -1 0 "" }{TEXT 461 74 "If the other t wo local exponent differences are equal to a rational number" }{TEXT 462 1 " " }{XPPEDIT 463 0 "1/k;" "6#*&\"\"\"F$%\"kG!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 464 48 "we have more transformations. They \+ have the form" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "``(1-z)-`>`;" "6#,&-%! G6#,&\"\"\"F(%\"zG!\"\"F(%\">GF*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-x )^m*(F[q](x)/F[p](x))^k;" "6#*&),&\"\"\"F&%\"xG!\"\"%\"mGF&)*&-&%\"FG6 #%\"qG6#F'F&-&F.6#%\"pG6#F'F(%\"kGF&" }{TEXT -1 1 " " }{TEXT 494 26 "a nd (by Hurwitz theorem) " }{XPPEDIT 18 0 "z-`>`*x^(p+q+1)*G[``](x);" "6#,&%\"zG\"\"\"*(%\">GF%)%\"xG,(%\"pGF%%\"qGF%F%F%F%-&%\"GG6#%!G6#F)F %!\"\"" }{TEXT 496 2 " ," }{TEXT -1 1 "\n" }{TEXT 495 6 "where " } {XPPEDIT 18 0 "F[p](x),F[q](x);" "6$-&%\"FG6#%\"pG6#%\"xG-&F%6#%\"qG6# F)" }{TEXT 497 27 " are polynomials of degree " }{XPPEDIT 18 0 "p,q;" "6$%\"pG%\"qG" }{TEXT 498 38 " respectively without multiple roots,\n " }{XPPEDIT 18 0 "G[``](x);" "6#-&%\"GG6#%!G6#%\"xG" }{TEXT -1 1 " " } {TEXT 499 44 "is a polynomial without multiple roots, and " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT 500 17 " does not divide " }{XPPEDIT 18 0 " m;" "6#%\"mG" }{TEXT 501 32 " .\nIt follows that the quotient " } {XPPEDIT 18 0 "F[q](x)/F[p](x);" "6#*&-&%\"FG6#%\"qG6#%\"xG\"\"\"-&F&6 #%\"pG6#F*!\"\"" }{TEXT -1 2 " " }{TEXT 502 28 "is the Pade aproximat ion of " }{XPPEDIT 18 0 "(1-x)^(-m/k);" "6#),&\"\"\"F%%\"xG!\"\",$*&% \"mGF%%\"kGF'F'" }{TEXT 503 11 " of degree " }{XPPEDIT 18 0 "[q*`/`*p] ;" "6#7#*(%\"qG\"\"\"%\"/GF&%\"pGF&" }{TEXT 504 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Hypergeom( [3/2,2],[4],c)=4/(4-3*x)*\nHyp ergeom([1/2,1],[2],x^3/(3*x-4)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%*HypergeomG6%7$#\"\"$\"\"#F*7#\"\"%%\"cG,$*(F,\"\"\",&F,F0*&F)F0%\" xGF0!\"\"F4-F%6%7$#F0F*F07#F**&F3F),&*&F)F0F3F0F0F,F4!\"#F0F0" }}} {EXCHG {PARA 259 "" 0 "" {TEXT -1 83 "These degenerate hypergeometric \+ functions can be tranformed to rational functions. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "Hypergeom([1/2,1],[2],-4*x^3*(x-1)^2*(x+2)/(3*x-2) ^2)=\n(2-3*x)/(1-x)^2/(x+2); checkseries(%,x, 8);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%*HypergeomG6%7$#\"\"\"\"\"#F)7#F*,$*,\"\"%F)%\"xG \"\"$,&F/F)F)!\"\"F*,&F/F)F*F)F),&*&F0F)F/F)F)F*F2!\"#F2*(,&F*F)*&F0F) F/F)F2F),&F)F)F/F2F6F3F2" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"xG\" \"\"\"\"!#!\"\"\"\"#\"\"$#!\"$\"\"%F.#!\"*\"\")\"\"&#!#B\"#;\"\"'#!#d \"#K\"\"(-%\"OG6#F&F1F$" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 40 "Note that the last identity is wrong at " }{XPPEDIT 552 0 "x = 1;" "6#/%\" xG\"\"\"" }{TEXT -1 4 " or " }{XPPEDIT 553 0 "x = -2;" "6#/%\"xG,$\"\" #!\"\"" }{TEXT -1 102 " . The reason is that analytic continuation of the left-hand side changes the branch when going from " }{XPPEDIT 554 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 220 " to these points. The st andard branch cuts of the Gauss hypergeometric function divides the co mplexplane into two isolated regions. We have other expression of the \+ left-hand side as a rational function around the points " }{XPPEDIT 555 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 1 " " }{TEXT 556 0 "" } {TEXT -1 4 "and " }{XPPEDIT 557 0 "x = -2;" "6#/%\"xG,$\"\"#!\"\"" } {TEXT -1 10 " , namely " }{XPPEDIT 558 0 "(3*x-2)/(x^3);" "6#*&,&*&\" \"$\"\"\"%\"xGF'F'\"\"#!\"\"F'*$F(F&F*" }{TEXT -1 2 " ." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 107 "Hypergeom([1/2,1],[2],-4*x^3*(x-1)^2*(x+2)/(3 *x-2)^2)=(3*x-2)/x^3;\ncheckseries(%,x=1), checkseries(%,x=-2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$#\"\"\"\"\"#F)7#F*,$ *,\"\"%F)%\"xG\"\"$,&F/F)F)!\"\"F*,&F/F)F*F)F),&*&F0F)F/F)F)F*F2!\"#F2 *&F4F)F/!\"$" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$/+/,&%\"xG\"\"\"F'!\" \"F'\"\"!!\"$\"\"#\"\")\"\"$!#:\"\"%\"#C\"\"&-%\"OG6#F'\"\"'F$/+1,&F&F 'F+F'F'F)#\"\"*F,F'#\"#:\"#;F+#\"#6F=F-#F<\"#KF/#\"#R\"$G\"F1F2F5F7" } }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 33 "Dihedral hypergeometric fun ctions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 466 66 "This is the cas e when two local exponent differences are equal to " }{XPPEDIT 18 0 "1 /2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 0 "" }{TEXT 467 34 ". \nBy a d egree two transformation " }{XPPEDIT 18 0 "(1/2, 1/2, p) < -``;" "6#26 %*&\"\"\"F&\"\"#!\"\"*&F&F&F'F(%\"pG,$%!GF(" }{XPPEDIT 18 0 "``(1,p,p) ;" "6#-%!G6%\"\"\"%\"pGF'" }{TEXT 468 137 " we can transform the hyper geometric equation to a degenerate hypergeometric equation with two si ngular points, as in the previous case. " }}{SECT 1 {PARA 263 "" 0 "" {TEXT 469 73 "Here are some hypergeometric identities and evaluation o f their solutions" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Hypergeom( [a/2,a/2+1/2],[a+1],4*x*(1-x))=Hypergeom( [a,a+1],[ a+1],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$,$*&\"\" #!\"\"%\"aG\"\"\"F-,&*&F*F+F,F-F-#F-F*F-7#,&F,F-F-F-,$*(\"\"%F-%\"xGF- ,&F-F-F6F+F-F--F%6%7$F,F2F1F6" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 9 "therefore" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Hypergeom([a/2,a/2+1/ 2],[a+1],z)=((1+sqrt(1-z))/2)^(-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%*HypergeomG6%7$,$*&\"\"#!\"\"%\"aG\"\"\"F-,&*&F*F+F,F-F-#F-F*F-7#, &F,F-F-F-%\"zG),&F0F-*&F*F+,&F-F-F3F+F0F-,$F,F+" }}}{EXCHG {PARA 259 " " 0 "" {TEXT -1 7 "Further" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Hyper geom( [a/2+1/2,a/2+1],[3/2],x^2/(2-x)^2)=\n(1-x/2)^(-a-1)*Hypergeom( [ a+1,1],[2],x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$,& *&\"\"#!\"\"%\"aG\"\"\"F-#F-F*F-,&*&F*F+F,F-F-F-F-7##\"\"$F**&%\"xGF*, &F*F-F5F+!\"#*&),&F-F-*&F*F+F5F-F+,&F,F+F-F+F--F%6%7$,&F,F-F-F-F-7#F*F 5F-" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 9 "therefore" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 161 "Hypergeom( [a/2+1/2,a/2+1],[3/2],z)=((1-sqrt( z))^(-a)-\n(1+sqrt(z))^(-a))/2/a/sqrt(z), a<>0;\nHypergeom([1/2,1],[3/ 2],z)=(log(1-sqrt(z))-log(1+sqrt(z)))/2/sqrt(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%*HypergeomG6%7$,&*&\"\"#!\"\"%\"aG\"\"\"F-#F-F*F-,&* &F*F+F,F-F-F-F-7##\"\"$F*%\"zG,$**F*F+,&),&F-F-*$F4F.F+,$F,F+F-),&F-F- F:F-F;F+F-F,F+F4#F+F*F-0F,\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% *HypergeomG6%7$#\"\"\"\"\"#F)7##\"\"$F*%\"zG,$*&F(F)*&,&-%#lnG6#,&F)F) *$F.F(!\"\"F)-F46#,&F)F)F7F)F8F)F.#F8F*F)F)" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 4 "Also" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Hypergeom([a /2,a/2+1/2],[1/2],z)=\n((1-sqrt(z))^(-a)+(1+sqrt(z))^(-a))/2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$,$*&\"\"#!\"\"%\"aG \"\"\"F-,&*&F*F+F,F-F-#F-F*F-7#F0%\"zG,&*&F*F+),&F-F-*$F2F0F+,$F,F+F-F -*&F*F+),&F-F-F7F-F8F-F-" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 470 9 "There are" }{TEXT -1 1 " " }{TEXT 471 16 "transformations " } {XPPEDIT 18 0 "(1/2, 1/2, p) < -``*(1, n*p, n*p);" "6#26%*&\"\"\"F&\" \"#!\"\"*&F&F&F'F(%\"pG,$*&%!GF&6%F&*&%\"nGF&F*F&*&F0F&F*F&F&F(" } {TEXT 472 11 " of degree " }{XPPEDIT 18 0 "2*n;" "6#*&\"\"#\"\"\"%\"nG F%" }{TEXT 473 142 ", they are compositions of the degree 2 transforma tions and transformations of the previous section.\nThe most interesti ng transformations are " }{XPPEDIT 18 0 "(1/2, 1/2, a) < -``(1/2,1/2,n *a);" "6#26%*&\"\"\"F&\"\"#!\"\"*&F&F&F'F(%\"aG,$-%!G6%*&F&F&F'F(*&F&F &F'F(*&%\"nGF&F*F&F(" }{TEXT -1 1 " " }{TEXT 474 251 ", to the same mo nodromy group. \nThey can be computed using the trick of expanding an \+ explicit solution of the transformed equation in a form that resembles (up to a radical factor) a solution of non-transformed equation. This works as follows:\nExpand " }{XPPEDIT 18 0 "(1+sqrt(x))^n;" "6#),&\" \"\"F%-%%sqrtG6#%\"xGF%%\"nG" }{TEXT -1 0 "" }{TEXT 476 4 " as " } {XPPEDIT 18 0 "theta[1](x)+theta[2](x)*sqrt(x);" "6#,&-&%&thetaG6#\"\" \"6#%\"xGF(*&-&F&6#\"\"#6#F*F(-%%sqrtG6#F*F(F(" }{TEXT 475 16 " with f unctions " }{XPPEDIT 18 0 "theta[1](x),theta[2](x);" "6$-&%&thetaG6#\" \"\"6#%\"xG-&F%6#\"\"#6#F)" }{TEXT -1 1 " " }{TEXT 477 12 "rational in " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 0 "" }{TEXT 478 30 "; then \+ the rational function " }{XPPEDIT 18 0 "x*theta[2](x)^2/(theta[1](x)^ 2);" "6#*(%\"xG\"\"\"*$-&%&thetaG6#\"\"#6#F$F+F%*$-&F)6#F%6#F$F+!\"\" " }{TEXT -1 2 " " }{TEXT 479 89 "defines the finite covering for the \+ pull-back. In particular, the (non-hypergeometric in " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT 480 11 ") solution " }{XPPEDIT 18 0 "(1-sqrt(z)) ^(-a);" "6#),&\"\"\"F%-%%sqrtG6#%\"zG!\"\",$%\"aGF*" }{TEXT -1 1 " " } {TEXT 481 43 "is transformed (up to a radical factor) to " }{XPPEDIT 18 0 "(1-sqrt(x*theta[2](x)^2/(theta[1](x)^2)))^(-a) = theta[1](x)^``( a)*(1-sqrt(x))^(-n*a);" "6#/),&\"\"\"F&-%%sqrtG6#*(%\"xGF&*$-&%&thetaG 6#\"\"#6#F+F1F&*$-&F/6#F&6#F+F1!\"\"F8,$%\"aGF8*&)-&F/6#F&6#F+-%!G6#F: F&),&F&F&-F(6#F+F8,$*&%\"nGF&F:F&F8F&" }{TEXT 482 2 ". " }}{SECT 1 {PARA 264 "" 0 "" {TEXT 483 56 "Explicitly, we have these closed form \+ identities here..." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "Hypergeom( [n*a/2,n*a/2+1/2],[1/2],x) = \n((1-sqrt(x ))^n/2+(1+sqrt(x))^n/2)^(-a)*\nHypergeom([a/2,a/2+1/2],[1/2],((1-sqrt( x))^n-(1+sqrt(x))^n)^2/\n((1-sqrt(x))^n+(1+sqrt(x))^n)^2 );" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 484 103 "Note that the argument of the \+ hypergeometric function on the right hand-side is a rational function \+ of " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT 487 16 ", and has value " } {XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT 485 4 " at " }{XPPEDIT 18 0 "x = 0 ;" "6#/%\"xG\"\"!" }{TEXT -1 0 "" }{TEXT 486 1 "." }{TEXT -1 0 "" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$,$*(\"\"#!\"\"%\"nG \"\"\"%\"aGF-F-,&#F-F*F-*(F*F+F,F-F.F-F-7#F0%\"xG*&),&*&F*F+),&F-F-*$F 3F0F+F,F-F-*&F*F+),&F-F-F:F-F,F-F-,$F.F+F--F%6%7$,$*&F*F+F.F-F-,&*&F*F +F.F-F-F0F-F2*&,&F8F-F " 0 "" {MPLTEXT 1 0 21 "checkseries(%, x,4 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+-%\"xG\"\"\"\"\"!,$**\"\"#!\"\"%\"nGF&%\"aGF&,&F&F&*& F,F&F-F&F&F&F&F&,$*.\"#CF+F,F&F-F&F.F&,&F*F&F/F&F&,&\"\"$F&F/F&F&F&F*, $*2\"$?(F+F,F&F-F&F.F&F3F&F4F&,&\"\"%F&F/F&F&,&\"\"&F&F/F&F&F&F5-%\"OG 6#F&F:F$" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 4 "Also" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 163 "Hypergeom( [n*a/2,n*a/2+1/2],[n*a+1],x) = \n( (1-sqrt(1-x))^n/2+(1+sqrt(1-x))^n/2)^(-a)*\nHypergeom([a/2,a/2+1/2],[a +1],4*x^n/\n((1-sqrt(1-x))^n+(1+sqrt(1-x))^n)^2 );" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/-%*HypergeomG6%7$,$*(\"\"#!\"\"%\"nG\"\"\"%\"aGF-F-, &*(F*F+F,F-F.F-F-#F-F*F-7#,&*&F,F-F.F-F-F-F-%\"xG*&),&*&F*F+),&F-F-*$, &F-F-F5F+F1F+F,F-F-*&F*F+),&F-F-F " 0 "" {MPLTEXT 1 0 135 "hypergeom([-1/2/k,1/2-1/2/k],[1-1/ k],1-x) = ((1+sqrt(x))/2)^(1/k);\nhypergeom([-1/2/k,-1/2-1/2/k],[-1/k] ,1-x) = ((1+sqrt(x))/2)^(1+1/k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %*hypergeomG6%7$,&#\"\"\"\"\"#F**&F*F**&F+F*%\"kGF*!\"\"F/,$*&F*F**&F+ F*F.F*F/F/7#,&F*F**&F*F*F.F/F/,&F*F*%\"xGF/),&F)F**&F+F/F7F)F*F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$,&#\"\"\"\"\"#!\"\"* &F*F**&F+F*%\"kGF*F,F,,$*&F*F**&F+F*F/F*F,F,7#,$*&F*F*F/F,F,,&F*F*%\"x GF,),&#F*F+F**&F+F,F7F:F*,&F*F*F5F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 510 29 "Here is a contiguity relation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "(1-1/k)*hypergeom([-1/2/k,-1/2-1/2/k],[1-1/k],1 -x)=\n(1+1/k)*hypergeom([-1/2/k,1/2-1/2/k],[1-1/k],1-x)-1/2/k*\nhyperg eom([-1/2/k,-1/2-1/2/k],[-1/k],1-x);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#/*&,&\"\"\"F&*&F&F&%\"kG!\"\"F)F&-%*hypergeomG6%7$,&#F&\"\"#F)*&F&F& *&F0F&F(F&F)F),$*&F&F&*&F0F&F(F&F)F)7#F%,&F&F&%\"xGF)F&,&*&,&F&F&F'F&F &-F+6%7$,&#F&F0F&*&F&F&*&F0F&F(F&F)F)F3F6F7F&F&*&#F&F0F&*&F(F)-F+6%F-7 #,$F'F)F7F&F&F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 511 13 "which implies" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "hypergeom([-1/ 2/k,-1/2-1/2/k],[1-1/k],1-x)=\n(k-sqrt(x))/(k-1)*((1+sqrt(x))/2)^(1/k) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$,&#\"\"\"\"\"#! \"\"*&F*F**&F+F*%\"kGF*F,F,,$*&F*F**&F+F*F/F*F,F,7#,&F*F**&F*F*F/F,F,, &F*F*%\"xGF,*(,&F/F**$F7#F*F+F,F*,&F/F*F*F,F,),&F;F**&F+F,F7F;F*F5F*" }}}{PARA 0 "" 0 "" {TEXT 516 0 "" }{TEXT -1 1 " " }{TEXT 517 11 "Now e xpand " }{XPPEDIT 18 0 "(k-sqrt(x))^k*(1+sqrt(x));" "6#*&),&%\"kG\"\" \"-%%sqrtG6#%\"xG!\"\"F&F',&F'F'-F)6#F+F'F'" }{TEXT -1 0 "" }{TEXT 513 4 " as " }{XPPEDIT 18 0 "theta[1](x)+theta[2](x)*sqrt(x);" "6#,&-& %&thetaG6#\"\"\"6#%\"xGF(*&-&F&6#\"\"#6#F*F(-%%sqrtG6#F*F(F(" }{TEXT 512 16 " with functions " }{XPPEDIT 18 0 "theta[1](x),theta[2](x);" "6 $-&%&thetaG6#\"\"\"6#%\"xG-&F%6#\"\"#6#F)" }{TEXT -1 1 " " }{TEXT 514 12 "rational in " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 0 "" }{TEXT 515 43 ";\nthe pull-back transformation is given by " }{XPPEDIT 18 0 " ``(1-z)-`>`;" "6#,&-%!G6#,&\"\"\"F(%\"zG!\"\"F(%\">GF*" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "1-x*theta[2](x)^2/(theta[1](x)^2);" "6#,&\"\"\"F$*(% \"xGF$*$-&%&thetaG6#\"\"#6#F&F,F$*$-&F*6#F$6#F&F,!\"\"F3" }{TEXT -1 1 " " }{TEXT 518 108 ".\nThis the the Klein pull-back transformation for a hypergeometric equation with local exponent differences " } {XPPEDIT 18 0 "``(1/2,3/2,1/k);" "6#-%!G6%*&\"\"\"F'\"\"#!\"\"*&\"\"$F 'F(F)*&F'F'%\"kGF)" }{TEXT 526 62 " . Here are two hypergeometric ide ntities in a general form::" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 270 "hypergeom([-1/2/k,-1/2-1/2/k],[1-1/k],1-x)=\n((k-s qrt(x))^k*(1+sqrt(x))/2+(k+sqrt(x))^k*(1-sqrt(x))/2)^(1/k)\n/(k-1)*hyp ergeom([-1/2/k,1/2-1/2/k],[1-1/k],1-\n((k-sqrt(x))^k*(1+sqrt(x))-(k+sq rt(x))^k*(1-sqrt(x)))^2/\n((k-sqrt(x))^k*(1+sqrt(x))+(k+sqrt(x))^k*(1- sqrt(x)))^2 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$,& #\"\"\"\"\"#!\"\"*&F*F**&F+F*%\"kGF*F,F,,$*&F*F**&F+F*F/F*F,F,7#,&F*F* *&F*F*F/F,F,,&F*F*%\"xGF,*(),&*(F+F,),&F/F**$F7#F*F+F,F/F*,&F*F*F>F*F* F**(F+F,),&F/F*F>F*F/F*,&F*F*F>F,F*F*F5F*,&F/F*F*F,F,-F%6%7$,&F?F**&F* F**&F+F*F/F*F,F,F0F3,&F*F**&,&*&F " 0 "" {MPLTEXT 1 0 259 "hypergeom([-1/2/k, -1/2-1/2/k],[-1/2],x)=\n((k-sqrt(x))^k*(1+sqrt(x))/2+(k+sqrt(x))^k*(1- sqrt(x))/2)^(1/k)\n/k*hypergeom([-1/2/k,1/2-1/2/k],[1/2],\n((k-sqrt(x) )^k*(1+sqrt(x))-(k+sqrt(x))^k*(1-sqrt(x)))^2/\n((k-sqrt(x))^k*(1+sqrt( x))+(k+sqrt(x))^k*(1-sqrt(x)))^2 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/-%*hypergeomG6%7$,&#\"\"\"\"\"#!\"\"*&F*F**&F+F*%\"kGF*F,F,,$*&F*F** &F+F*F/F*F,F,7##F,F+%\"xG*(),&*(F+F,),&F/F**$F5#F*F+F,F/F*,&F*F*FF*F**&F@F*FBF*F,F+,&FMF*FNF*!\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "subs( (k^k)^(1/k)=k, chec kseries(%,x,5));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+/%\"xG\"\"\"\"\" !,$*(\"\"#!\"\",&%\"kGF&F&F&F&F-!\"#F+F&,$*,\"\")F+F,F&,&F-F&F&F+F&,&F &F+*&F*F&F-F&F&F&F-!\"%F+F*,$*0\"$W\"F+F,F&F2F&F3F&,&*&\"\"$F&F-F&F&F& F+F&,&F&F+*&\"\"%F&F-F&F&F&F-!\"'F+F;,$*4\"%gdF+F,F&F2F&F3F&F9F&F-%\"OG6#F&FE,.F& F&**F*F+F,F&F-F.F%F&F+*.F1F+F-F5F,F&F2F&F3F&F%F*F+*2F8F+F-F?F2F&F,F&F9 F&F3F&FF+-FK6#*$)F% FEF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 34 "Algebraic hypergeometric functions" }} {PARA 259 "" 0 "" {TEXT -1 250 "Algebraic hypergeometric functions for m a very classical subject. They were classified by Schwartz. If we ig nore\nhypergeometric functions with finite cyclic or dihedral monodrom y groups (see subcases of previous two cases),\nwe may have tetrahedra l ( " }{XPPEDIT 519 0 "A[4];" "6#&%\"AG6#\"\"%" }{TEXT -1 17 " ), octa hedral ( " }{XPPEDIT 520 0 "S[4];" "6#&%\"SG6#\"\"%" }{TEXT -1 20 " ) \+ or icosahedral ( " }{XPPEDIT 521 0 "A[5];" "6#&%\"AG6#\"\"&" }{TEXT -1 313 " ) monodromy groups. There are 2 tetrahedral, 2 octahedral and 10 icosahedral Schwartz types of algebraic hypergeometric functions. \+ Hypergeometric functions of the same Schwartz type are contiguous to e ach other (up to fractional-linear transformations). \nHypergeometric \+ equations with local exponent differences " }{XPPEDIT 522 0 "``(1/2,1/ 3,1/3),``(1/2,1/3,1/4),``(1/2,1/3,1/5);" "6%-%!G6%*&\"\"\"F'\"\"#!\"\" *&F'F'\"\"$F)*&F'F'F+F)-F$6%*&F'F'F(F)*&F'F'F+F)*&F'F'\"\"%F)-F$6%*&F' F'F(F)*&F'F'F+F)*&F'F'\"\"&F)" }{TEXT -1 12 " are called " }{TEXT 523 68 "standard Gauss hypergeomeometric functions with algebraic solution s." }{TEXT -1 907 " They have the tetrahedral, octahedral, icosahedral monodromy group respectively. The known theorem of Klein implies that any hypergeometric equations with algebraic solutions (only) is a pul l-back transformation of the standard hypergeometric equation with the same monodromy group. The most interesting transformations of algebra ic hypergeometric functions are those implied by the Klein's theorem. \+ Those transformations can be computed by using a data base of explicit expressions of some algebraic Gauss hypergeometric functions, contigu ous relations to compute such explicit expressions for any algebraic G auss functions, and suitable identification of quotients of two algebr aic hypergeometric solutions.\nThe mentioned explicit expressions have the following meaning: we can pull-back a hypergeometric equation und er consideration to a hypergeometric equation with a cyclic monodromy \+ group (so called " }{TEXT 524 17 "Darboux pull-back" }{TEXT -1 264 "); then hypergeometric solutions are transformed to radical functions, w hich are indeed very explicit.\nIn the case of hypergeometric equation s with finite dihedral monodromy group, a Darboux pull-back transforma tion is a simple transformation of degree 2, such as " }{XPPEDIT 525 0 "z < -``*x^2;" "6#2%\"zG,$*&%!G\"\"\"*$%\"xG\"\"#F(!\"\"" }{TEXT -1 257 ", and we considered the mentioned\nmethod (for computing Klein's morphism) in simplified form in the previous case.\nWe give here simp lest Darboux expressions for other algebraic hypergeometric functions, and give one example of Klein's morphism computation." }}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 28 "Tetrahedral monodromy group " }}{SECT 1 {PARA 262 "" 0 "" {TEXT 302 13 "Schwartz type" }{TEXT -1 1 " " } {XPPEDIT 18 0 "` `(1/2,1/3,1/3);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F '\"\"$F)*&F'F'F+F)" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 264 "Here we p resent a Darboux expression for the easiest standard solution with tet rahedral monodromy group, and a similar expression for a contiguous fu nction. With contiguous relations one can compute such form for any ot her contiguous Gauss hypergeometric function." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "presenthyperids(1); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6$/-%*hypergeomG6%7$#!\"\"\"#7#\"\"\"\"\"%7##\"\"#\"\"$,$**F-F)%\"xGF ,,&F4F,F-F,F1,&*&F0F,F4F,F,F,F)!\"$F,*&F,F,*$),&F,F,*&F0F,F4F,F)#F,F-F ,F)/-F%6%7$F(#\"\"&F-7##FDF1F2*(,&F,F,F4F,F,,&F,F,*&F-F)F4F,F,!\"#F<#F )F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "checkseries( [%][1], x, 8 );" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+5%\"xG\" \"\"\"\"!#F&\"\"#F&#\"\"&\"\")F)#\"#:\"#;\"\"$#\"$&>\"$G\"\"\"%#\"$j' \"$c#F+#\"%TY\"%C5\"\"'#\"&vl\"\"%[?\"\"(-%\"OG6#F&F,F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 527 130 "A similar expression for the second hypergeometric solution of the standard tetrahedral equati on,\nand for its contiguous function." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "presenthyperids(2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$/-%*hypergeomG6%7$#\"\"\"\"\"%#\"\"(\"#77##F*\"\"$,$**F *!\"\"%\"xGF),&F4F)F*F)F0,&*&\"\"#F)F4F)F)F)F3!\"$F)*&,&F)F)*&F8F)F4F) F3#F0F*,&F)F)*&F*F3F4F)F)F3/-F%6%7$#!\"&F-F(7##F)F0F1*&,&F)F)*(\"\"&F) F8F3F4F)F)F)F;#FEF*" }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT 301 13 "Schwa rtz type" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/3,1/3,2/3);" "6#-%\"~G 6%*&\"\"\"F'\"\"$!\"\"*&F'F'F(F)*&\"\"#F'F(F)" }}{EXCHG {PARA 0 "" 0 " " {TEXT 528 286 "Here we present explicit expressions for hypergeometr ic solutions of a representative hypergeometric equation of the second tetrahedral type, and instances of contiguous evaluations. The contig uous evaluations are put in the form which is more conveninet for usin g contiguous relations." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "presenth yperidz(3); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6$/-%*hypergeomG6%7$#!\" \"\"\"'#\"\"\"\"\"#7##F-\"\"$*(%\"xGF,,&F2F,F-F,F0,&*&F-F,F2F,F,F,F,! \"$*&F,F,*$F4#F,F-F)/-F%6%7$F+#\"\"&F*F.F1*(F4F-,&F,F,F2F)!\"#F$F," }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "presenthyperidz(4);\n" }} {PARA 12 "" 1 "" {XPPMATH 20 "6$/-%*hypergeomG6%7$#\"\"\"\"\"'#\"\"&F* 7##\"\"%\"\"$*(%\"xGF),&F2F)\"\"#F)F0,&*&F4F)F2F)F)F)F)!\"$*(F5#F)F4,& F)F)F2F)#F)F0,&F)F)*&F4!\"\"F2F)F)F>/-F%6%7$#F>F*F(7#F;F1*(FF$F )" }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT 300 16 "A Klein morphism" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 529 85 "As an example, we conside r a hypergeometric equation with local exponent differences " } {XPPEDIT 18 0 "``(4/3,4/3,2/3);" "6#-%!G6%*&\"\"%\"\"\"\"\"$!\"\"*&F'F (F)F**&\"\"#F(F)F*" }{TEXT -1 0 "" }{TEXT 530 66 ".\nDarboux pull-back transformations for this equation is given by " }{XPPEDIT 18 0 "X = x *(x+2)^3/((2*x+1)^3);" "6#/%\"XG*(%\"xG\"\"\"*$,&F&F'\"\"#F'\"\"$F'*$, &*&F*F'F&F'F'F'F'F+!\"\"" }{TEXT -1 1 " " }{TEXT 531 65 ",\nsuch a tra nsformation for the standard tetrahedral equation is " }{XPPEDIT 18 0 "Z = z*(z+4)^3/4/((2*z-1)^3);" "6#/%\"ZG**%\"zG\"\"\"*$,&F&F'\"\"%F'\" \"$F'F*!\"\"*$,&*&\"\"#F'F&F'F'F'F,F+F," }{TEXT -1 0 "" }{TEXT 532 2 " ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "XZ:= [numer(X-x*(x+2)^ 3/(2*x+1)^3),numer(Z-z*(z+4)^3/4/(2*z-1)^3)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 533 0 "" }{TEXT -1 0 "" }{TEXT 534 83 "A quoti ent of two standard tetrahedral solutions is (by the expicit formulas \+ above)" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "Eq1:= Z^(1 /3)*hypergeom( [1/4,7/12], [2/3], Z)/\nhypergeom( [1/4,-1/12], [2/3], \+ Z )=(-16*z)^(1/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Eq1G/*(%\"ZG# \"\"\"\"\"$-%*hypergeomG6%7$#F)\"\"%#\"\"(\"#77##\"\"#F*F'F)-F,6%7$#! \"\"F3F/F4F'F;*$),$*&\"#;F)%\"zGF)F;F(F)" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 134 "To compute the corresponding (up to a radical factor) hy pergeometric solutions of the chosen equation,\nwe recall contiguous r elations:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Hypergeom([-7/6,-1/2] ,[-1/3],X)=(1-X)/2*Hypergeom([5/6, 1/2],[2/3],X)+(1-3*X)/2*Hypergeom([ -1/6, 1/2],[2/3],X);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG 6%7$#!\"(\"\"'#!\"\"\"\"#7##F,\"\"$%\"XG,&*&#\"\"\"F-F5*&,&F5F5F1F,F5- F%6%7$#\"\"&F*F47##F-F0F1F5F5F5*&F4F5*&,&F5F5*&F0F5F1F5F,F5-F%6%7$#F,F *F4F=F1F5F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "Hypergeom ([1/6, 5/6],[7/3],X)=16/21/X*Hypergeom([1/6, -1/6],[1/3],X)+8/21*(3*X- 2)/X*Hypergeom([1/6, 5/6],[4/3],X);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/-%*HypergeomG6%7$#\"\"\"\"\"'#\"\"&F*7##\"\"(\"\"$%\"XG,&*&#\"#;\"#@ F)*&F1!\"\"-F%6%7$F(#F8F*7##F)F0F1F)F)F)*&#\"\")F6F)*(,&*&F0F)F1F)F)\" \"#F8F)F1F8-F%6%F'7##\"\"%F0F1F)F)F)" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 44 "Together with above evaluations, they imply " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 174 "Eq2:= X^(4/3)*Hypergeom([1/6,5/6],[7/3],X)/ hypergeom([-7/6,-1/2],[-1/3],X)=32/21*x^(1/3)*(x+1)^(1/3)*(x-1)^2/(2*x +1)*\n((3*X-2)*(2*x+1)+x+2)/((1-3*X)*(1-x)^2+(1-X)*(1+2*x)^2);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%$Eq2G/*(%\"XG#\"\"%\"\"$-%*Hypergeom G6%7$#\"\"\"\"\"'#\"\"&F17##\"\"(F*F'F0-%*hypergeomG6%7$#!\"(F1#!\"\" \"\"#7##F>F*F'F>,$*2\"#KF0\"#@F>%\"xG#F0F*,&F0F0FFF0FG,&FFF0F0F>F?,&*& F?F0FFF0F0F0F0F>,(*&,&*&F*F0F'F0F0F?F>F0FJF0F0FFF0F?F0F0,&*&,&F0F0*&F* F0F'F0F>F0),&F0F0FFF>F?F0F0*&,&F0F0F'F>F0)FJF?F0F0F>F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 535 141 "The two quotients of hypergeometric fucntions can be compared up to a scalar factor.\nTo find the scalar \+ factor, we compare both quotients at " }{XPPEDIT 18 0 "x = -2,X = 0,Z \+ = 0,z = -4;" "6&/%\"xG,$\"\"#!\"\"/%\"XG\"\"!/%\"ZGF*/%\"zG,$\"\"%F'" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "subs(x=-2, X=0,Z=0,z=-4,op(2,Eq1)=c*op(2,Eq2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)\"#k#\"\"\"\"\"$F(,$*,\"#KF(\"#@!\"\"%\"cGF(!\"#F'F.F'F." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 536 17 "and conclude that" }{TEXT -1 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "Eqq:= z=2*x*(x+1)*(x-1)^6/(2*x+ 1)^3*\n((3*X-2)*(2*x+1)+x+2)^3/((1-3*X)*(1-x)^2+(1-X)*(1+2*x)^2)^3;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EqqG/%\"zG,$*0\"\"#\"\"\"%\"xGF*,& F*F*F+F*F*,&F+F*F*!\"\"\"\"',&*&F)F*F+F*F*F*F*!\"$,(*&,&*&\"\"$F*%\"XG F*F*F)F.F*F0F*F*F+F*F)F*F7,&*&,&F*F**&F7F*F8F*F.F*),&F*F*F+F.F)F*F**&, &F*F*F8F.F*)F0F)F*F*F2F*" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 16 "Now we eliminate" }{TEXT 537 1 " " }{XPPEDIT 538 0 "x,z;" "6$%\"xG%\"zG" }{TEXT -1 20 " from the relations" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "T:= numer(z-op(2,Eqq)):\nresultant(XZ[2],T,z):\nresultant(XZ[1],%, x):\nre:= factor(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#reG,$*(\"jn C-)4)[#Ri8v!p%[$4`+C]s#\\@@53B6-Dl`L\"\"\"),R*&\"%'4%F(%\"ZGF(F(*(\"&; g)F(F-F(%\"XGF(!\"\"*(\"'!GL)F(F-F()F0\"\"#F(F(*(\"(%eJ\\F(F-F()F0\"\" $F(F1*&\"&Wq$F()F0\"\"%F(F(*(\"))[&))>F(F-F(F;B\"F(F-F(FHF(F( *&\")#p![;F()F0\"\"(F(F1*(\"*)RQe>F(FNF(F-F(F1*&\")#\\/S%F()F0\"\")F(F (*(\"*^'=,BF(FTF(F-F(F(*&\")W<&)zF()F0\"\"*F(F1*(\"*gm.&>F(FZF(F-F(F1* &\")s4j**F()F0\"#5F(F(*(\"*\"HrK6F(FjnF(F-F(F(*&\")s>K%)F()F0\"#6F(F1* (\")9w]SF(F`oF(F-F(F1*&\");WHYF()F0\"#7F(F(*(\"(p7v'F(FfoF(F-F(F(*&\") [.)[\"F()F0\"#8F(F1*&\"(kd7#F()F0\"#9F(F(F=F(),&F0F(F(F1\"#CF(F(" }}} {EXCHG {PARA 259 "" 0 "" {TEXT -1 31 "There is only one solution for \+ " }{XPPEDIT 539 0 "Z;" "6#%\"ZG" }{TEXT -1 2 " !" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "so:= map(factor, [solve(re,Z)] );" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%#soG7&,$*,\"$3\"\"\"\"%\"XG\"\"%,(*&\"#FF))F*\"\"#F )F)*&F.F)F*F)!\"\"\"\"(F)\"\"$,&F*F)F)F2F+,,*&\"$*=F))F*F+F)F)*&\"$y$F ))F*F4F)F2*&\"$,$F)F/F)F)*&\"$7\"F)F*F)F2\"#;F)!\"$F2F&F&F&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 540 76 "Here is an example of hypergeometric identity for this Klein transformation:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "hypergeom( [-1/2,-7/6], [-1/3], X )=\n(1-7 *X+301/16*X^2-189/8*X^3+189/16*X^4)^(1/4)*\nhypergeom( [1/4,-1/12], [2 /3], so[1] );\ncheckseries( %, X, 8 ); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#!\"(\"\"'#!\"\"\"\"#7##F,\"\"$%\"XG*&),,\"\" \"F5*&\"\"(F5F1F5F,*&#\"$,$\"#;F5*$)F1F-F5F5F5*&#\"$*=\"\")F5*$)F1F0F5 F5F,*&#F@F;F5*$)F1\"\"%F5F5F5#F5FHF5-F%6%7$#F,\"#7FI7##F-F0,$*,\"$3\"F 5F1FH,(*&\"#FF5F=F5F5*&FVF5F1F5F,F7F5F0,&F1F5F5F,FH,,*&F@F5FGF5F5*&\"$ y$F5FCF5F,*&F:F5F=F5F5*&\"$7\"F5F1F5F,F;F5!\"$F,F5" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/+5%\"XG\"\"\"\"\"!#!\"(\"\"%F&#\"\"(\"#k\"\"##F,\"$c #\"\"$#\"$&Q\"&oF$F*#\"$L)\"'s58\"\"&#\"%6#)\"(_r4#\"\"'#\"&6?#\"(3')Q )F,-%\"OG6#F&\"\")F$" }}}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 26 "Octa hedral monodromy group" }}{SECT 1 {PARA 262 "" 0 "" {TEXT 303 13 "Schw artz type" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/2,1/3,1/4);" "6#-%\"~ G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)*&F'F'\"\"%F)" }{TEXT -1 0 "" }} {EXCHG {PARA 259 "" 0 "" {TEXT -1 609 "From now on we present only Dar boux evaluations of hypergeometric solutions of the simplest hypergeo metric equations of each Schwartz type. Contiguous evaluations can be \+ obtained by differentiating the given identities, and (consequently) b y using contiguous relations like above. \nConvenient contiguous evalu ations can be viewed by calling programmed procedures\n \"presenthyper ids\" or \"presenthyperidz\". Klein morphisms can be computed as in th e given tetrahedral example, with chosen standard choises of quotients of hypergeometric functions to compare, and evaluation points to dete rmine a scalar multiple." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "present hyperid(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#!\" \"\"#C#\"\"(F*7##\"\"$\"\"%,$**\"$3\"\"\"\"%\"xGF4,&F5F4F4F)F0,(*$)F5 \"\"#F4F4*&\"#9F4F5F4F4F4F4!\"$F4*&F4F4*$)F7#F4\"\")F4F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "presenthyperid(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#\"\"&\"#C#\"#8F*7##F)\"\"%,$** \"$3\"\"\"\"%\"xGF3,&F4F3F3!\"\"F/,(*$)F4\"\"#F3F3*&\"#9F3F4F3F3F3F3! \"$F3*&F7#F)\"\"),&F3F3F4F6F6" }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT 304 13 "Schwartz type" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(2/3,1/4,1/4 );" "6#-%\"~G6%*&\"\"#\"\"\"\"\"$!\"\"*&F(F(\"\"%F**&F(F(F,F*" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "presenthyperid(7); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#!\"\"\"#7#\"\"( F*7##\"\"$\"\"%,$*,\"#F\"\"\"\"\"#F)%\"xGF4,&F4F4F6F4F0,(*$)F6F5F4F4*& F0F4F6F4F4F4F4!\"$F4*&,&F4F4*&F5F)F6F4F4#F4F0F8#F)F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "presenthyperid(8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#\"\"\"\"\"'#\"\"&F*7##F,\"\"%,$*,\"# FF)\"\"#!\"\"%\"xGF),&F)F)F5F)F/,(*$)F5F3F)F)*&F/F)F5F)F)F)F)!\"$F)*(, &*&F3F)F5F)F)F)F)#F)F/F7#F)F3F6F4" }}}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 27 "Icosahedral monodromy group" }}{SECT 1 {PARA 262 "" 0 "" {TEXT 305 14 "Schwartz types" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/2, 1/3,1/5);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)*&F'F'\"\"&F)" }{TEXT -1 2 " " }{TEXT 306 4 "and " }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/2,1/3,2/5);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)*&F(F'\" \"&F)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "presenthyperid(9); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG ,$**\"%G<\"\"\"%\"xGF),(*$)F*\"\"#F)F)*&\"#6F)F*F)!\"\"F)F1\"\"&,,*$)F *\"\"%F)F)*&\"$G#F))F*\"\"$F)F)*&\"$%\\F)F-F)F)*&F8F)F*F)F1F)F)!\"$F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#!\"\"\"#g#\"#>F *7##\"\"%\"\"&%$PsiG*&\"\"\"F3*$),,*$)%\"xGF/F3F3*&\"$G#F3)F9\"\"$F3F3 *&\"$%\\F3)F9\"\"#F3F3*&F;F3F9F3F)F3F3#F3\"#?F3F)" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 73 "We check the series in this way ('subsdarboux ' is a global variable, the " }{XPPEDIT 649 0 "Psi;" "6#%$PsiG" } {TEXT -1 14 "-substitution)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "chec kseries( subs(subsdarboux,%), x );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# /+1%\"xG\"\"\"\"\"!#\"#d\"\"&F&#\"&(\\L\"#D\"\"##\")BY$e#\"$D\"\"\"$# \",%z#4RA#\"$D'\"\"%#\"0=pJgno,\"\"&Dc\"F*-%\"OG6#F&\"\"'F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "presenthyperid(10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$** \"%G<\"\"\"%\"xGF),(*$)F*\"\"#F)F)*&\"#6F)F*F)!\"\"F)F1\"\"&,,*$)F*\" \"%F)F)*&\"$G#F))F*\"\"$F)F)*&\"$%\\F)F-F)F)*&F8F)F*F)F1F)F)!\"$F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#\"#6\"#g#\"#JF*7## \"\"'\"\"&%$PsiG*&,,*$)%\"xG\"\"%\"\"\"F8*&\"$G#F8)F6\"\"$F8F8*&\"$%\\ F8)F6\"\"#F8F8*&F:F8F6F8!\"\"F8F8#F)\"#?,(F8F8*&F)F8F6F8F8*$F?F8FBFB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "presenthyperid(11);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$** \"%G<\"\"\"%\"xGF),(*$)F*\"\"#F)F)*&\"#6F)F*F)!\"\"F)F1\"\"&,,*$)F*\" \"%F)F)*&\"$G#F))F*\"\"$F)F)*&\"$%\\F)F-F)F)*&F8F)F*F)F1F)F)!\"$F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#!\"(\"#g#\"#8F*7## \"\"$\"\"&%$PsiG*&,&\"\"\"F4*&\"\"(F4%\"xGF4!\"\"F4,,*$)F7\"\"%F4F4*& \"$G#F4)F7F/F4F4*&\"$%\\F4)F7\"\"#F4F4*&F>F4F7F4F8F4F4#F)\"#?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "presenthyperid(12);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$** \"%G<\"\"\"%\"xGF),(*$)F*\"\"#F)F)*&\"#6F)F*F)!\"\"F)F1\"\"&,,*$)F*\" \"%F)F)*&\"$G#F))F*\"\"$F)F)*&\"$%\\F)F-F)F)*&F8F)F*F)F1F)F)!\"$F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#\"#<\"#g#\"#PF*7## \"\"(\"\"&%$PsiG*(,&\"\"\"F4*&F/!\"\"%\"xGF4F4F4,,*$)F7\"\"%F4F4*&\"$G #F4)F7\"\"$F4F4*&\"$%\\F4)F7\"\"#F4F4*&F=F4F7F4F6F4F4#F)\"#?,(F4F4*&\" #6F4F7F4F4*$FBF4F6!\"#" }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT 307 13 "Sc hwartz type" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/2,1/5,2/5);" "6#-% \"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"&F)*&F(F'F+F)" }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "presenthyperid(13);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$*, \"#k\"\"\"%\"xGF),(*$)F*\"\"#F)F)F*!\"\"F)F/\"\"&,&F,F)F)F/F/,(F,F)*& \"\"%F)F*F)F)F)F/!\"&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hyperge omG6%7$#!\"\"\"#?#\"\"(F*7##\"\"%\"\"&%$PsiG*(,&\"\"\"F4%\"xGF4F+,&F4F 4F5F)F(,(F4F4*&F/F4F5F4F)*$)F5\"\"#F4F)#F)F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "presenthyperid(14);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$*,\"#k\"\"\"%\"xGF) ,(*$)F*\"\"#F)F)F*!\"\"F)F/\"\"&,&F,F)F)F/F/,(F,F)*&\"\"%F)F*F)F)F)F/! \"&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#\"\"$\"#?# \"#6F*7##\"\"'\"\"&%$PsiG**,&\"\"\"F4%\"xGF4F(,&F4F4F5!\"\"F+,(F4F4*& \"\"%F4F5F4F7*$)F5\"\"#F4F7#F)F:,(F4F4F5F4F;F7F7" }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT 309 14 "Schwartz types" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/3,1/3,2/5);" "6#-%\"~G6%*&\"\"\"F'\"\"$!\"\"*&F'F'F(F)*&\" \"#F'\"\"&F)" }{TEXT -1 2 " " }{TEXT 308 3 "and" }{TEXT -1 2 " " } {XPPEDIT 18 0 "` `(1/3,2/3,1/5);" "6#-%\"~G6%*&\"\"\"F'\"\"$!\"\"*&\" \"#F'F(F)*&F'F'\"\"&F)" }{TEXT -1 0 "" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 320 "For the remaining Schwartz icosahedral types the simples t Darboux transformationsare not between projective lines, but from a \+ projective line to a genus 1 curve ! In these cases Darboux evaluation s are radical functions on the genus 1 curve. To check the given ident ities, one may look at the power series expensions in " }{XPPEDIT 541 0 "sqrt(x);" "6#-%%sqrtG6#%\"xG" }{TEXT -1 65 ". In computation of Kle in morphisms, one would have to eliminate " }{XPPEDIT 542 0 "z,x;" "6$ %\"zG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 543 0 "y;" "6#%\"yG" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "H15:= presenthyperid(15) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darboux~curve~is:G/*$)%\"yG \"\"#\"\"\"*&%\"xGF),(F)F)*&\"#LF)F+F)F)*&\"\"*F))F+F(F)!\"\"F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$*, \"$W\"\"\"\"%\"yGF),(F)F)*&\"#LF)%\"xGF)F)*&\"\"*F))F.\"\"#F)!\"\"F2,( F)F)*&F0F)F*F)F3*&\"#aF)F.F)F)F),,F)F)*&\"#@F)F*F)F)*&\"$<\"F)F.F)F3*( F0F)F.F)F*F)F)*&\"$M#F)F1F)F3!\"$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$H15G/-%*hypergeomG6%7$#!\"\"\"#I#\"\"$\"#57##F.\"\"&%$PsiG*&,(\"\" \"F6*&\"\"*F6%\"yGF6F+*&\"#aF6%\"xGF6F6#F6F,,,F6F6*&\"#@F6F9F6F6*&\"$< \"F6F " 0 "" {MPLTEXT 1 0 44 "checkseries( subs(subsdarboux, H15), x, 3 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.\"\"\"F%*(\"#7F%\"\"&!\"\"% \"xG#F%\"\"#F)*(\"$F*F%\"#DF)F*F%F%*(\"&)*Q)F%\"$D\"F)F*#\"\"$F,F)*(\" (\\%z%)F%\"$D'F)F*F,F%-%\"OG6#*$)F*#F(F,F%F%,0F%F%*(F'F%F(F)F*F+F)*(F. F%F/F)F*F%F%*(F1F%F2F)F*F3F)*(F6F%F7F)F*F,F%*(\"+h(*z^!*F%\"&]7$F)F*F= F)-F96#*$)F*F4F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "prese nthyperid(16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darboux~curve~ is:G/*$)%\"yG\"\"#\"\"\"*&%\"xGF),(F)F)*&\"#LF)F+F)F)*&\"\"*F))F+F(F)! \"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/ %$PsiG,$*,\"$W\"\"\"\"%\"yGF),(F)F)*&\"#LF)%\"xGF)F)*&\"\"*F))F.\"\"#F )!\"\"F2,(F)F)*&F0F)F*F)F3*&\"#aF)F.F)F)F),,F)F)*&\"#@F)F*F)F)*&\"$<\" F)F.F)F3*(F0F)F.F)F*F)F)*&\"$M#F)F1F)F3!\"$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#\"#6\"#I#\"\"(\"#57##F,\"\"&%$PsiG*( ,,\"\"\"F4*&\"#@F4%\"yGF4F4*&\"$<\"F4%\"xGF4!\"\"*(\"\"*F4F:F4F7F4F4*& \"$M#F4)F:\"\"#F4F;#F)F-,(F4F4*&F=F4F7F4F;*&\"#aF4F:F4F4#!#6F*,(F4F4*& \"#LF4F:F4F4*&F=F4F@F4F;F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "presenthyperid(17);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darbo ux~curve~is:G/*$)%\"yG\"\"#\"\"\"*&%\"xGF),(F)F)*&\"#LF)F+F)F)*&\"\"*F ))F+F(F)!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morph ism~is:G/%$PsiG,$*,\"$W\"\"\"\"%\"yGF),(F)F)*&\"#LF)%\"xGF)F)*&\"\"*F) )F.\"\"#F)!\"\"F2,(F)F)*&F0F)F*F)F3*&\"#aF)F.F)F)F),,F)F)*&\"#@F)F*F)F )*&\"$<\"F)F.F)F3*(F0F)F.F)F*F)F)*&\"$M#F)F1F)F3!\"$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#!\"\"\"#5#\"#<\"#I7##\"\"%\"\"& %$PsiG*&,(\"\"\"F5*&\"\"*F5%\"yGF5F)*&\"#aF5%\"xGF5F5#\"#8F-,,F5F5*&\" #@F5F8F5F5*&\"$<\"F5F;F5F)*(F7F5F;F5F8F5F5*&\"$M#F5)F;\"\"#F5F)#!\"$F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "presenthyperid(18);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darboux~curve~is:G/*$)%\"yG\"\"# \"\"\"*&%\"xGF),(F)F)*&\"#LF)F+F)F)*&\"\"*F))F+F(F)!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$*,\"$W\" \"\"\"%\"yGF),(F)F)*&\"#LF)%\"xGF)F)*&\"\"*F))F.\"\"#F)!\"\"F2,(F)F)*& F0F)F*F)F3*&\"#aF)F.F)F)F),,F)F)*&\"#@F)F*F)F)*&\"$<\"F)F.F)F3*(F0F)F. F)F*F)F)*&\"$M#F)F1F)F3!\"$F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*h ypergeomG6%7$#\"\"\"\"#5#\"#B\"#I7##\"\"'\"\"&%$PsiG*,,,F)F)*&\"#@F)% \"yGF)F)*&\"$<\"F)%\"xGF)!\"\"*(\"\"*F)F:F)F7F)F)*&\"$M#F))F:\"\"#F)F; #\"\"$F*,(F)F)*&F=F)F7F)F;*&\"#aF)F:F)F)#\"\"(F-,&F7F)*&F1F)F:F)F)F)F7 F;,&F)F)*&F=F)F:F)F)F;" }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT 310 14 "Sc hwartz types" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(2/3,1/5,1/5);" "6#-% \"~G6%*&\"\"#\"\"\"\"\"$!\"\"*&F(F(\"\"&F**&F(F(F,F*" }{TEXT -1 2 " \+ " }{TEXT 311 3 "and" }{TEXT -1 2 " " }{XPPEDIT 18 0 "` `(1/3,2/5,3/5) ;" "6#-%\"~G6%*&\"\"\"F'\"\"$!\"\"*&\"\"#F'\"\"&F)*&F(F'F,F)" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "presenthyperid(19) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darboux~curve~is:G/*$)%\"yG \"\"#\"\"\"*&%\"xGF),(F)F)*&\"\"&F)F+F)F)*&F.F))F+F(F)!\"\"F)" }} {PARA 12 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$*0 \"$K%\"\"\"%\"xGF),*\"\"&F)*&\"\"(F)%\"yGF)!\"\"*&\"#XF)F*F)F0*&F,F))F *\"\"#F)F0F,,,F)F)*&\"#]F)F*F)F)*&\"$D\"F))F/F5F)F0*(\"$]%F)F*F)F/F)F) *&\"$+&F)F4F)F0F),&*&F,F)F/F)F)*&\"#dF)F*F)F)F0,*F,F)*&\"#=F)F/F)F)*& \"#!)F)F*F)F0*&F,F)F4F)F)!\"&,,F)F)*&F8F)F*F)F)*&F:F)F;F)F0*(F=F)F*F)F /F)F0*&F?F)F4F)F0F0F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*hypergeom G6%7$#!\"\"\"#I#\"\"\"\"\"'7##\"\"%\"\"&%$PsiG*(,(F,F,*(\"\"$F,F1F)%\" yGF,F)*(\"#MF,F1F)%\"xGF,F)F+,(F,F,*&F6F,F7F,F,*&\"#?F,F:F,F)#F)F-,,F, F,*&\"#]F,F:F,F,*&\"$D\"F,)F7\"\"#F,F)*(\"$]%F,F:F,F7F,F)*&\"$+&F,)F:F FF,F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "presenthyperid(2 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darboux~curve~is:G/*$)%\" yG\"\"#\"\"\"*&%\"xGF),(F)F)*&\"\"&F)F+F)F)*&F.F))F+F(F)!\"\"F)" }} {PARA 12 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$*0 \"$K%\"\"\"%\"xGF),*\"\"&F)*&\"\"(F)%\"yGF)!\"\"*&\"#XF)F*F)F0*&F,F))F *\"\"#F)F0F,,,F)F)*&\"#]F)F*F)F)*&\"$D\"F))F/F5F)F0*(\"$]%F)F*F)F/F)F) *&\"$+&F)F4F)F0F),&*&F,F)F/F)F)*&\"#dF)F*F)F)F0,*F,F)*&\"#=F)F/F)F)*& \"#!)F)F*F)F0*&F,F)F4F)F)!\"&,,F)F)*&F8F)F*F)F)*&F:F)F;F)F0*(F=F)F*F)F /F)F0*&F?F)F4F)F0F0F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*hypergeom G6%7$#\"\"\"\"\"'#\"#6\"#I7##F*\"\"&%$PsiG*0,,F)F)*&\"#]F)%\"xGF)F)*& \"$D\"F))%\"yG\"\"#F)!\"\"*(\"$]%F)F6F)F:F)F<*&\"$+&F))F6F;F)F " 0 "" {MPLTEXT 1 0 19 "present hyperid(21);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darboux~curve~is :G/*$)%\"yG\"\"#\"\"\"*&%\"xGF),(F)F)*&\"\"&F)F+F)F)*&F.F))F+F(F)!\"\" F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$Ps iG,$*0\"$K%\"\"\"%\"xGF),*\"\"&F)*&\"\"(F)%\"yGF)!\"\"*&\"#XF)F*F)F0*& F,F))F*\"\"#F)F0F,,,F)F)*&\"#]F)F*F)F)*&\"$D\"F))F/F5F)F0*(\"$]%F)F*F) F/F)F)*&\"$+&F)F4F)F0F),&*&F,F)F/F)F)*&\"#dF)F*F)F)F0,*F,F)*&\"#=F)F/F )F)*&\"#!)F)F*F)F0*&F,F)F4F)F)!\"&,,F)F)*&F8F)F*F)F)*&F:F)F;F)F0*(F=F) F*F)F/F)F0*&F?F)F4F)F0F0F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*hype rgeomG6%7$#!\"\"\"\"'#\"#8\"#I7##\"\"$\"\"&%$PsiG*,,,\"\"\"F5*&\"#]F5% \"xGF5F5*&\"$D\"F5)%\"yG\"\"#F5F)*(\"$]%F5F8F5F " 0 "" {MPLTEXT 1 0 19 "presenthyperid (22);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darboux~curve~is:G/*$)% \"yG\"\"#\"\"\"*&%\"xGF),(F)F)*&\"\"&F)F+F)F)*&F.F))F+F(F)!\"\"F)" }} {PARA 12 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$*0 \"$K%\"\"\"%\"xGF),*\"\"&F)*&\"\"(F)%\"yGF)!\"\"*&\"#XF)F*F)F0*&F,F))F *\"\"#F)F0F,,,F)F)*&\"#]F)F*F)F)*&\"$D\"F))F/F5F)F0*(\"$]%F)F*F)F/F)F) *&\"$+&F)F4F)F0F),&*&F,F)F/F)F)*&\"#dF)F*F)F)F0,*F,F)*&\"#=F)F/F)F)*& \"#!)F)F*F)F0*&F,F)F4F)F)!\"&,,F)F)*&F8F)F*F)F)*&F:F)F;F)F0*(F=F)F*F)F /F)F0*&F?F)F4F)F0F0F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*hypergeom G6%7$#\"\"(\"#I#\"\"&\"\"'7##F)F,%$PsiG*.,*\"\"\"F3*&#\"#=F,F3%\"yGF3F 3*&\"#;F3%\"xGF3!\"\"*$)F:\"\"#F3F3#F)F-,&F3F3*&\"#DF;F:F3F3F+,(F3F3*& F,F3F7F3F3*&\"#5F3F:F3F3F3,,F3F3*&\"#]F3F:F3F3*&\"$D\"F3)F7F>F3F;*(\"$ ]%F3F:F3F7F3F;*&\"$+&F3F=F3F;F(,*F3F3*&#F)F,F3F7F3F;*&\"\"*F3F:F3F;F " 0 "" {MPLTEXT 1 0 19 "presenthyperid(23);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darboux~curve~is:G/*$)%\"yG\" \"#\"\"\"*(%\"xGF),&F)F)F+F)F),&F)F)*&\"#;F)F+F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$*0\"#a\"\"\",&% \"yGF)*&\"\"&F)%\"xGF)F)\"\"$,(F)F)*&\"\"#F)F+F)!\"\"*&\"\"'F)F.F)F)F- ,&F+F)*&F-F)F.F)F3!\"#,&F)F)*&\"\"%F)F.F)F)F3,&F)F)*&F;F)F.F)F3F3,(F)F )*&F2F)F+F)F3*&\"#9F)F.F)F3!\"&F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%*hypergeomG6%7$#!\"\"\"#:#\"\")F*7##\"\"%\"\"&%$PsiG*,,&%\"yG\"\"\"* &F0F5%\"xGF5F5#F5\"\"',&F5F5*&F/F5F7F5F5F+F7#F5F*,(F5F5*&\"\"#F5F4F5F) *&\"#9F5F7F5F)#F)\"\"$,&F4F5*&FCF5F7F5F)#!\"$\"#5" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "presenthyperid(24);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darboux~curve~is:G/*$)%\"yG\"\"#\"\"\"*(%\"xGF),& F)F)F+F)F),&F)F)*&\"#;F)F+F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9 The~Darboux~morphism~is:G/%$PsiG,$*0\"#a\"\"\",&%\"yGF)*&\"\"&F)%\"xGF )F)\"\"$,(F)F)*&\"\"#F)F+F)!\"\"*&\"\"'F)F.F)F)F-,&F+F)*&F-F)F.F)F3!\" #,&F)F)*&\"\"%F)F.F)F)F3,&F)F)*&F;F)F.F)F3F3,(F)F)*&F2F)F+F)F3*&\"#9F) F.F)F3!\"&F3" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#\" \"#\"#:#\"#6F*7##\"\"'\"\"&%$PsiG*2,(\"\"\"F4*&F)F4%\"yGF4!\"\"*&\"#9F 4%\"xGF4F7#F)\"\"$,(F4F4*&F)F4F6F4F7*&F/F4F:F4F4F7,&F6F4*&F0F4F:F4F4#F 4F/,&F6F4*&F " 0 "" {MPLTEXT 1 0 19 "presenthyperid(25) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darboux~curve~is:G/*$)%\"yG \"\"#\"\"\"*&%\"xGF),(F)F)F+F)*$)F+F(F)!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$*.\"#;\"\"\"%\"yGF) ,(F)F)%\"xGF)*$)F,\"\"#F)!\"\"F/,&F)F)F*F0F/,(F)F)F*F)*&F/F)F,F)F)F0,( F)F)F*F)*&F/F)F,F)F0!\"&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hype rgeomG6%7$#!\"\"\"#5#\"\"(F*7##\"\"%\"\"&%$PsiG**,(\"\"\"F4%\"yGF)*&\" \"#F4%\"xGF4F4#F4\"#:,&F4F4F5F)#\"\"$F0,(F4F4F5F4*&F7F4F8F4F4#!\"(\"#I ,(F4F4F5F4*&F7F4F8F4F)#F)F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "presenthyperid(26);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darbo ux~curve~is:G/*$)%\"yG\"\"#\"\"\"*&%\"xGF),(F)F)F+F)*$)F+F(F)!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG,$ *.\"#;\"\"\"%\"yGF),(F)F)%\"xGF)*$)F,\"\"#F)!\"\"F/,&F)F)F*F0F/,(F)F)F *F)*&F/F)F,F)F)F0,(F)F)F*F)*&F/F)F,F)F0!\"&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#\"\"\"\"#5#\"\"*F*7##\"\"'\"\"&%$Psi G*0,(%\"yGF)*&\"\"#F)%\"xGF)F)*$)F7F6F)F)F),&F)F)F4F)F(,&F)F)F4!\"\"# \"\"$F*F4F<,(F)F)F4F)*&F6F)F7F)F)#!\"#\"#:,(F)F)F4F<*&F6F)F7F)F)#F<\"# I,(F)F)F4F)*&F6F)F7F)F<#F " 0 "" {MPLTEXT 1 0 19 "presenthyperid(27);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Dar boux~curve~is:G/*$)%\"yG\"\"#\"\"\"*&%\"xGF),(F)F)F+F)*$)F+F(F)!\"\"F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morphism~is:G/%$PsiG ,$*.\"#;\"\"\"%\"yGF),(F)F)%\"xGF)*$)F,\"\"#F)!\"\"F/,&F)F)F*F0F/,(F)F )F*F)*&F/F)F,F)F)F0,(F)F)F*F)*&F/F)F,F)F0!\"&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#!\"\"\"#5#\"\"$F*7##F,\"\"&%$PsiG**, (\"\"\"F3%\"yGF3*&\"\"#F3%\"xGF3F3#F3\"#I,&F3F3F4F)#F3F/,(F3F3F4F)*&F6 F3F7F3F3#F6\"#:,(F3F3F4F3*&F6F3F7F3F)#F)F6" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "presenthyperid(28);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6The~Darboux~curve~is:G/*$)%\"yG\"\"#\"\"\"*&%\"xGF),(F)F)F+F)* $)F+F(F)!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9The~Darboux~morph ism~is:G/%$PsiG,$*.\"#;\"\"\"%\"yGF),(F)F)%\"xGF)*$)F,\"\"#F)!\"\"F/,& F)F)F*F0F/,(F)F)F*F)*&F/F)F,F)F)F0,(F)F)F*F)*&F/F)F,F)F0!\"&F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#\"\"$\"#5#\"\"(F*7# #F,\"\"&%$PsiG*.,(\"\"\"F3%\"yGF3*&\"\"#F3%\"xGF3F3#F,\"#I,&F3F3F4F3#F 3F/,(F3F3F4F3*&F6F3F7F3!\"\"#F)F6,&F3F3F4F>#!\"#F/,(F3F3F4F>*&F6F3F7F3 F3#F>\"#:,(F3F3F7F3*$)F7F6F3F>F>" }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT 315 16 "A Klein morphism" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 544 37 "Sorry, this example is not ready yet." }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 21 "Other transformations" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 546 218 "Transformations of algebraic Gauss hypergeometric fu nctions are special cases of classical transformations, except when th e transformed equation (up to fractional-linear transformations) has l ocal exponent differences " }{XPPEDIT 18 0 "``(1/k,1/m,1/n);" "6#-%!G6 %*&\"\"\"F'%\"kG!\"\"*&F'F'%\"mGF)*&F'F'%\"nGF)" }{TEXT -1 1 " " } {TEXT 547 8 ", where " }{XPPEDIT 18 0 "k,m,n;" "6%%\"kG%\"mG%\"nG" } {TEXT -1 1 " " }{TEXT 548 26 "are positive integers and " }{XPPEDIT 18 0 "1 < 1/k+1/m+1/n;" "6#2\"\"\",(*&F$F$%\"kG!\"\"F$*&F$F$%\"mGF(F$* &F$F$%\"nGF(F$" }{TEXT -1 0 "" }{TEXT 549 354 " . The most interesting transformations are Klein's pull-back transformations to hypergeometr ic equations with the same monodromy group.\nTransformations to hyperg eometric equations with a smaller monodromy group are either special c ases of classical transformations, or compositions of transformations \+ of smaller degree, except a degree 5 transformation " }{XPPEDIT 18 0 " ``(1/2,1/3,1/5) < -``(1/2,1/3,1/3);" "6#2-%!G6%*&\"\"\"F(\"\"#!\"\"*&F (F(\"\"$F**&F(F(\"\"&F*,$-F%6%*&F(F(F)F**&F(F(F,F**&F(F(F,F*F*" } {TEXT 550 65 " . Here is an example of a corresponding hypergeometric \+ identity:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "hypergeom( [1/ 4,-1/12], [2/3], x) = \n(1+(7-33*\\sqrt(-15))/128*x)^(1/12)*hypergeom \n([11/60,-1/60],[2/3],\n50*(5+3*\\sqrt(-15))*x*(1024*x-781-171*sqrt(- 15))^3/\n(128*x+7+33*sqrt(-15))^5 );" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#/-%*hypergeomG6%7$#!\"\"\"#7#\"\"\"\"\"%7##\"\"#\"\"$%\"xG*&),&F,F,* (\"$G\"F),&\"\"(F,*&^#!#LF,\"#:#F,F0F,F,F2F,F,#F,F*F,-F%6%7$#F)\"#g#\" #6FDF.,$*,\"#]F,,&\"\"&F,*&^#F1F,F=F>F,F,F2F,,(*&\"%C5F,F2F,F,\"$\"yF) *&^#!$r\"F,F=F>F,F1,(*&F7F,F2F,F,F9F,*&^#\"#LF,F=F>F,!\"&F,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "checkseries( %, x);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"xG\"\"\"\"\"!#!\"\"\"#KF&#!#6\"% C5\"\"##!$f(\"'s58\"\"$#!&&RJ\"(3')Q)\"\"%#!'.nr\"*caVo#\"\"&-%\"OG6#F &\"\"'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 33 "Hypergeometric elliptic integrals" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 545 109 "Here we consider transfo rmations of hypergeometric equations (and functions) with local expone nt differences " }{XPPEDIT 18 0 "``(1/k,1/m,1/n);" "6#-%!G6%*&\"\"\"F' %\"kG!\"\"*&F'F'%\"mGF)*&F'F'%\"nGF)" }{TEXT -1 1 " " }{TEXT 559 8 ", \+ where " }{XPPEDIT 18 0 "k,m,n;" "6%%\"kG%\"mG%\"nG" }{TEXT 561 32 " ar e positive integers uch that " }{XPPEDIT 18 0 "1/k+1/m+1/n = 1;" "6#/, (*&\"\"\"F&%\"kG!\"\"F&*&F&F&%\"mGF(F&*&F&F&%\"nGF(F&F&" }{TEXT -1 1 " " }{TEXT 560 50 ". Instances of such hypergeometric equations are " }{XPPEDIT 18 0 "diff(y(z),`$`(z,2))+((1-1/k)/z-(1-1/m)/(z-1))*diff(y(z ),z);" "6#,&-%%diffG6$-%\"yG6#%\"zG-%\"$G6$F*\"\"#\"\"\"*&,&*&,&F/F/*& F/F/%\"kG!\"\"F6F/F*F6F/*&,&F/F/*&F/F/%\"mGF6F6F/,&F*F/F/F6F6F6F/-F%6$ -F(6#F*F*F/F/" }{XPPEDIT 18 0 "`` = 0;" "6#/%!G\"\"!" }{TEXT -1 1 " " }{TEXT 562 149 ", so some of their solutions are constants or radical \+ functions, others are integrals (by the method of variation of constan ts). Those integrals are " }{TEXT 563 20 "elliptic integrals,\n" } {TEXT 564 100 "i.e., they represent integrals on genus 1 curves. In ou r case we deal with three different curves: " }{XPPEDIT 18 0 "y^2 = x ^3-x;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"F)!\"\"" }{TEXT -1 0 "" } {TEXT 566 2 " ," }{XPPEDIT 18 0 "y^2 = x^3-1;" "6#/*$%\"yG\"\"#,&*$%\" xG\"\"$\"\"\"F+!\"\"" }{TEXT 565 3 " , " }{XPPEDIT 18 0 "x^3+y^3 = 1; " "6#/,&*$%\"xG\"\"$\"\"\"*$%\"yGF'F(F(" }{TEXT 567 101 " (the last tw o are isomorphic curves). Together with the chord-and-tangent group st ructure, they are " }{TEXT 568 15 "elliptic curves" }{TEXT 569 629 ". \+ The interesting hypergeometric solutions can be represented by integra ls of holomorphic differentials. Endomorphisms of those elliptic curve s transform them to holomorphic integrals again,\nso the integrand cha nges by a constant multiple. This eventually implies that the endomor phisms correspond (up to the automorphisms) presicely to transformatio ns of the corresponding hypergeometric equations into themselves (up t o the fractional-linear transformations). Other transformations of th ose hypergeometric functions are either special cases of clasical tran sformations, or are compositions of smaller degree transformations. " }}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 27 "Transformations for l.e.d. " }{XPPEDIT 18 0 "` `(1/2,1/4,1/4);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F' F'\"\"%F)*&F'F'F+F)" }}{EXCHG {PARA 0 "" 0 "" {TEXT 580 96 "A represen tative hypergeometric function for these local exponent differences is the following: " }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "z^(1/4)*Hypergeom([1/2,1/4],[5/4],z)=1/4*\n'int( t^(-3/4)*(1-t)^(-1/2 ), t=0..z )';\ncheckseries(%, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ *&)%\"zG#\"\"\"\"\"%F(-%*HypergeomG6%7$#F(\"\"#F'7##\"\"&F)F&F(,$*&F'F (-%$intG6$*&F(F(*&)%\"tG#\"\"$F)F(,&F(F(F;!\"\"#F(F/F?/F;;\"\"!F&F(F( " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*$)%\"zG#\"\"\"\"\"%F)F)*&\"#5! \"\"F'#\"\"&F*F)*&\"#CF-F'#\"\"*F*F)*(F/F)\"$3#F-F'#\"#8F*F)*(\"#NF)\" %w@F-F'#\"#`;" "6#,&%\"tG\"\"\"%\">G!\"\"" }{TEXT 582 1 " " }{XPPEDIT 258 0 "1/(x^2);" "6#*&\"\"\"F$*$%\"xG\"\"#!\"\"" }{TEXT -1 73 " of th e integration variable we obviously get a holomorphic integral on " } {XPPEDIT 259 0 "y^2 = x^3-x;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"F)! \"\"" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "z^(1/4)*Hyp ergeom([1/2,1/4],[5/4],z)=1/2*\n'int( 1/sqrt(x^3-x), x=1/sqrt(z)..infi nity )';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"zG#\"\"\"\"\"%F(-%* HypergeomG6%7$#F(\"\"#F'7##\"\"&F)F&F(,$*&F.F(-%$intG6$*&F(F(-%%sqrtG6 #,&*$)%\"xG\"\"$F(F(F?!\"\"FA/F?;*&F(F(-F:6#F&FA%)infinityGF(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 279 3 "If " }{XPPEDIT 18 0 "``(x, y)-`>`;" "6#,&-%!G6$%\"xG%\"yG\"\"\"%\">G!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(phi(x,y),psi(x,y));" "6#-%!G6$-%$phiG6$%\"xG%\"yG-%$ psiG6$F)F*" }{TEXT -1 2 " " }{TEXT 280 41 "is an endomorphism of the \+ elliptic curve " }{XPPEDIT 18 0 "y^2 = x^3-x;" "6#/*$%\"yG\"\"#,&*$%\" xG\"\"$\"\"\"F)!\"\"" }{TEXT -1 2 ", " }{TEXT 570 225 "then the integr and differential is transformed to a holomorphic differential again, s o it is changed by a constant multiple.\n The upper integration bound \+ is unchanged, the change of the lower bound gives the transformation\n " }{XPPEDIT 18 0 "z-`>`;" "6#,&%\"zG\"\"\"%\">G!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "1/(phi(1/sqrt(z))^2);" "6#*&\"\"\"F$*$-%$phiG6#*&F$F $-%%sqrtG6#%\"zG!\"\"\"\"#F." }{TEXT -1 2 " " }{TEXT 571 49 " for the variable of the hypergeometric equation." }{TEXT -1 1 "\n" }{TEXT 572 26 "Note that endomorphisms of" }{TEXT -1 2 " " }{XPPEDIT 281 0 " y^2 = x^3-x;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"F)!\"\"" }{TEXT -1 1 " " }{TEXT 573 26 "form a ring isomorphic to " }{TEXT -1 1 " " } {TEXT 282 1 "Z" }{XPPEDIT 283 0 "[i];" "6#7#%\"iG" }{TEXT -1 2 ".\n" } {TEXT 574 145 "Since automorhisms imply trivial transformations, the g roup of pull-transformations of this hypergeometric equation into itse lf is isomorphic to " }{TEXT 575 1 "Z" }{XPPEDIT 257 0 "[i]^`*`;" "6#) 7#%\"iG%\"*G" }{TEXT -1 0 "" }{TEXT 576 0 "" }{XPPEDIT 577 0 "`/`(1,i, -1,-i);" "6#-%\"/G6&\"\"\"%\"iG,$F&!\"\",$F'F)" }{TEXT -1 1 " " } {TEXT 578 138 ". The degree of hypergeometric transformation is equal \+ to the degree of the endomorphism, or the norm of a corresponding Gaus sian integer." }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 284 30 " This is the multiplication by " }{TEXT -1 1 "2" }{TEXT 581 13 " endomo rphism" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "\{x =(x^2+1)^2/4/x/(x^2-1), y=(x^2+1)*(x^4-6*x^2+1)/8/x/y/(x^2-1)\};" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG,$**\"\"%!\"\",&\"\"\"F+*$)F% \"\"#F+F+F.F%F),&F,F+F+F)F)F+/%\"yG,$*.\"\")F)F*F+,(*$)F%F(F+F+*&\"\"' F+F-F+F)F+F+F+F%F)F1F)F/F)F+" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 22 "Indeed maps the curve " }{XPPEDIT 256 0 "y^2 = x^3-x;" "6#/*$%\"yG\" \"#,&*$%\"xG\"\"$\"\"\"F)!\"\"" }{TEXT -1 13 " into itself:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "factor( subs(%,-y^2+x^3-x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*4\"#k!\"\",&\"\"\"F(*$)%\"xG\"\"#F(F(F,,( F)F(*&F,F(F+F(F&F(F&F,,(F)F(*&F,F(F+F(F(F(F&F,,(*$)%\"yGF,F(F&*$)F+\" \"$F(F(F+F&F(F+!\"$F4!\"#,&F+F(F(F&F8,&F(F(F+F(F8F&" }}}{PARA 259 "" 0 "" {TEXT -1 54 "And this is the corresponding hypergeometric identit y:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "Hypergeom([1/2,1/4],[5 /4],z)=sqrt(1-z)/(1+z)*\nHypergeom([1/2,1/4],[5/4],16*z*(z-1)^2/(z+1)^ 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$#\"\"\"\"\"## F)\"\"%7##\"\"&F,%\"zG*(,&F)F)F0!\"\"F(,&F)F)F0F)F3-F%6%F'F-,$**\"#;F) F0F),&F0F)F)F3F*F4!\"%F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "checkseries(%,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"zG\"\" \"\"\"!#F&\"#5F&#F&\"#C\"\"##\"\"&\"$3#\"\"$#\"#N\"%w@\"\"%#F0\"$c#F.- %\"OG6#F&\"\"'F$" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 285 30 "This is th e multiplication by " }{XPPEDIT 18 0 "1+2*i;" "6#,&\"\"\"F$*&\"\"#F$% \"iGF$F$" }{TEXT 286 13 " endomorphism" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "\{x=x*(x^2-1-2*I)^2/((1+2*I)*x^2-1)^2, y= y*(x^2-1-2*I)*(x^4+(2+8*I)*x^2+1)/((1+2*I)*x^2-1)^3\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG*(F%\"\"\",&*$)F%\"\"#F'F'^$!\"\"!\"#F'F+, &*&^$F'F+F'F*F'F'F'F-F./%\"yG**F3F'F(F',(*$)F%\"\"%F'F'*&^$F+\"\")F'F* F'F'F'F'F'F/!\"$" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 22 "Indeed maps the curve " }{XPPEDIT 579 0 "y^2 = x^3-x;" "6#/*$%\"yG\"\"#,&*$%\"xG \"\"$\"\"\"F)!\"\"" }{TEXT -1 13 " into itself:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "factor( subs(%,-y^2+x^3-x) );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*.^$\"$<\"!#W\"\"\",(*$)%\"yG\"\"#F'!\"\"*$)%\"xG\"\"$F 'F'F0F-F',&*$)F0F,F'F'^$F-!\"#F'F,,*F3F'*&F,F'F0F'F'*&^#F6F'F0F'F'F'F' F,,*F3F'*&F,F'F0F'F-*&^#F,F'F0F'F'F'F'F,,&*&\"\"&F'F4F'F'^$F-F,F'!\"' " }}}{PARA 0 "" 0 "" {TEXT 287 52 "And this is a corresponding hyperge ometric identity." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "Hypergeom([1/2,1/4],[5/4],z)=(1-z/(1+2*I))/(1-(1+2*I)*z)*\nHy pergeom([1/2,1/4],[5/4],z*(z-1-2*I)^4/((1+2*I)*z-1)^4);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$#\"\"\"\"\"##F)\"\"%7##\"\"&F, %\"zG*(,&F)F)*&^$#!\"\"F/#F*F/F)F0F)F)F),&F)F)*&^$F6!\"#F)F0F)F)F6-F%6 %F'F-*(F0F),&F0F)F:F)F,,&*&^$F)F*F)F0F)F)F)F6!\"%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map(series,%,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"zG\"\"\"\"\"!#F&\"#5F&#F&\"#C\"\"##\"\"&\"$3#\"\" $#\"#N\"%w@\"\"%#F0\"$c#F.-%\"OG6#F&\"\"'F$" }}}}}{SECT 1 {PARA 260 " " 0 "" {TEXT -1 27 "Transformations for l.e.d. " }{XPPEDIT 18 0 "` `(1 /2,1/3,1/6);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)*&F'F'\"\"'F )" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 583 96 "A representat ive hypergeometric function for these local exponent differences is th e following: " }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "z^( 1/6)*Hypergeom([1/2,1/6],[7/6],z)=1/6*\n'int'( t^(-5/6)*(1-t)^(-1/2), \+ t=0..z );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"zG#\"\"\"\"\"'F( -%*HypergeomG6%7$#F(\"\"#F'7##\"\"(F)F&F(,$*&F'F(-%$intG6$*&F(F(*&)%\" tG#\"\"&F)F(,&F(F(F;!\"\"#F(F/F?/F;;\"\"!F&F(F(" }}}{EXCHG {PARA 259 " " 0 "" {TEXT -1 14 "By the change " }{XPPEDIT 256 0 "t-`>`;" "6#,&%\"t G\"\"\"%\">G!\"\"" }{TEXT 584 1 " " }{XPPEDIT 257 0 "1/(x^3);" "6#*&\" \"\"F$*$%\"xG\"\"$!\"\"" }{TEXT -1 73 " of the integration variable w e obviously get a holomorphic integral on " }{XPPEDIT 258 0 "y^2 = x^3 -1;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"F+!\"\"" }{TEXT -1 1 "." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "z^(1/6)*Hypergeom([1/2,1/6],[7/6],z )=1/2*\n'int'( 1/sqrt(x^3-1), x=1/z^(1/3)..infinity );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"zG#\"\"\"\"\"'F(-%*HypergeomG6%7$#F(\"\"#F' 7##\"\"(F)F&F(,$*&F.F(-%$intG6$*&F(F(*$,&*$)%\"xG\"\"$F(F(F(!\"\"#F(F/ F?/F=;*&F(F(*$)F&#F(F>F(F?%)infinityGF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 288 3 "If " }{XPPEDIT 18 0 "``(x,y)-`>`;" "6#,&-%!G6$% \"xG%\"yG\"\"\"%\">G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(phi(x,y) ,psi(x,y));" "6#-%!G6$-%$phiG6$%\"xG%\"yG-%$psiG6$F)F*" }{TEXT -1 2 " \+ " }{TEXT 289 42 "is an endomorphism of the elliptic curve " } {XPPEDIT 18 0 "y^2 = x^3-1;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"F+!\" \"" }{TEXT 613 1 "," }{TEXT -1 1 "\n" }{TEXT 290 6 "then " }{XPPEDIT 18 0 "z-`>`;" "6#,&%\"zG\"\"\"%\">G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/(phi(z^(-1/3))^3);" "6#*&\"\"\"F$*$-%$phiG6#)%\"zG,$*&F$F$\"\" $!\"\"F.F-F." }{TEXT -1 2 " " }{TEXT 291 48 "gives a transformation o f the elliptic integral." }}{PARA 259 "" 0 "" {TEXT -1 18 "Endomorphis ms of " }{XPPEDIT 260 0 "y^2 = x^3-1;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$ \"\"\"F+!\"\"" }{TEXT -1 29 " form a ring isomorphic to " }{TEXT 292 1 "Z" }{XPPEDIT 262 0 "[omega];" "6#7#%&omegaG" }{TEXT -1 8 ", whe re " }{XPPEDIT 585 0 "omega = exp(2*Pi*i/3);" "6#/%&omegaG-%$expG6#** \"\"#\"\"\"%#PiGF*%\"iGF*\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 586 77 "The group of transformations of this hy pergeometric equation into itself is " }{TEXT 587 1 "Z" }{XPPEDIT 18 0 "[omega]^`*`;" "6#)7#%&omegaG%\"*G" }{TEXT -1 0 "" }{TEXT 590 1 "/" }{TEXT -1 0 "" }{TEXT 589 0 "" }{TEXT -1 0 "" }{XPPEDIT 18 0 "``(1,ome ga,omega^2,-1.-omega,-omega^2);" "6#-%!G6'\"\"\"%&omegaG*$F'\"\"#,&-%& FloatG6$F&\"\"!!\"\"F'F/,$*$F'F)F/" }{TEXT -1 0 "" }{TEXT 588 1 "." }} {SECT 1 {PARA 262 "" 0 "" {TEXT 591 18 "Multiplication by " }{TEXT 592 1 "3" }{TEXT 593 13 " endomorphism" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "Hypergeom([1/2,1/6],[7/6],z)=(1-4*z)/sqrt(1+96*z+48* z^2-64*z^3)*\nHypergeom([1/2,1/6],[7/6],-729*z*(4*z-1)^6/(64*z^3-48*z^ 2-96*z-1)^3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$#\" \"\"\"\"##F)\"\"'7##\"\"(F,%\"zG*(,&F)F)*&\"\"%F)F0F)!\"\"F),*F)F)*&\" #'*F)F0F)F)*&\"#[F))F0F*F)F)*&\"#kF))F0\"\"$F)F5#F5F*-F%6%F'F-,$**\"$H (F)F0F),&*&F4F)F0F)F)F)F5F,,**&F=F)F>F)F)*&F:F)F;F)F5*&F8F)F0F)F5F)F5! \"$F5F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "checkseries(%,z) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"zG\"\"\"\"\"!#F&\"#9F&#\"\" $\"$/\"\"\"##\"\"&\"$/$F+#\"\"(\"$S'\"\"%#\"#j\"%OzF/-%\"OG6#F&\"\"'F$ " }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT -1 0 "" }{TEXT 594 18 "Multiplic ation by " }{XPPEDIT 595 0 "3+omega;" "6#,&\"\"$\"\"\"%&omegaGF%" } {TEXT -1 1 " " }{TEXT 596 12 "endomorphsim" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 181 "Hypergeom([1/2,1/6],[7/6],z)=(1-4*z/(3*omega+1))/ \nsqrt(1-(44+48*omega)*z+(48*omega+16)*z^2)*\nHypergeom([1/2,1/6],[7/6 ],z*(4*z-3*omega-1)^6/\n((48*omega+16)*z^2-(44+48*omega)*z+1)^3);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$#\"\"\"\"\"##F)\"\"' 7##\"\"(F,%\"zG*(,&F)F)*(\"\"%F)F0F),&*&\"\"$F)%&omegaGF)F)F)F)!\"\"F9 F),(*&,&*&\"#[F)F8F)F)\"#;F)F))F0F*F)F)*&,&\"#WF)*&F>F)F8F)F)F)F0F)F9F )F)#F9F*-F%6%F'F-*(F0F),(*&F4F)F0F)F)*&F7F)F8F)F9F)F9F,F:!\"$F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "checkseries(%,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"zG\"\"\"\"\"!#F&\"#9F&#\"\"$\"$/\"\"\" ##\"\"&\"$/$F+#\"\"(\"$S'\"\"%#\"#j\"%OzF/-%\"OG6#F&\"\"'F$" }}}}} {SECT 1 {PARA 260 "" 0 "" {TEXT -1 27 "Transformations for l.e.d. " } {XPPEDIT 18 0 "` `(1/3,1/3,1/3);" "6#-%\"~G6%*&\"\"\"F'\"\"$!\"\"*&F'F 'F(F)*&F'F'F(F)" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 597 96 "A representative hypergeometric function for these local exponent dif ferences is the following: " }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "z^(1/6)*Hypergeom([1/3,2/3],[4/3],z)=1/6*\n'int( t^(- 2/3)*(1-t)^(-2/3), t=0..z )';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)% \"zG#\"\"\"\"\"'F(-%*HypergeomG6%7$#F(\"\"$#\"\"#F/7##\"\"%F/F&F(,$*&F 'F(-%$intG6$*&F(F(*&)%\"tG#F1F/F(),&F(F(F=!\"\"#F1F/F(FA/F=;\"\"!F&F(F (" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 14 "By the change " }{XPPEDIT 256 0 "t-`>`;" "6#,&%\"tG\"\"\"%\">G!\"\"" }{TEXT 598 1 " " }{XPPEDIT 257 0 "1/(x^3);" "6#*&\"\"\"F$*$%\"xG\"\"$!\"\"" }{TEXT -1 73 " of th e integration variable we obviously get a holomorphic integral on " } {XPPEDIT 258 0 "x^3+y^3 = 1;" "6#/,&*$%\"xG\"\"$\"\"\"*$%\"yGF'F(F(" } {TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "z^(1/6)*Hypergeom( [1/3,2/3],[4/3],z)=1/2*\n'int( 1/(x^3-1)^(2/3), x=z^(-1/3)..infinity ) ';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"zG#\"\"\"\"\"'F(-%*Hyperg eomG6%7$#F(\"\"$#\"\"#F/7##\"\"%F/F&F(,$*&#F(F1F(-%$intG6$*&F(F(*$),&* $)%\"xGF/F(F(F(!\"\"#F1F/F(FB/FA;*&F(F(*$)F&#F(F/F(FB%)infinityGF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 599 53 "This is an integral of a holomorphic differential on " }{XPPEDIT 18 0 "x^3+y^3 = 1;" "6#/,&* $%\"xG\"\"$\"\"\"*$%\"yGF'F(F(" }{TEXT -1 0 "" }{TEXT 600 67 ".\nWe ma ke this curve an eliptic curve by taking the infinity point " } {XPPEDIT 18 0 "``(x,y,1) = ``(1,-1,0);" "6#/-%!G6%%\"xG%\"yG\"\"\"-F%6 %F),$F)!\"\"\"\"!" }{TEXT -1 1 " " }{TEXT 601 149 "as the origin point .\nThen there is a chord-and-tangent group structure on the curve, whe re to compute the additive opposite one has to use the lines " } {XPPEDIT 18 0 "x+y = const;" "6#/,&%\"xG\"\"\"%\"yGF&%&constG" }{TEXT 602 27 " instead of vertical lines." }}{SECT 1 {PARA 262 "" 0 "" {TEXT 297 19 "The eliptic curves " }{TEXT -1 1 " " }{TEXT 293 0 "" } {XPPEDIT 294 0 "y^2 = x^3+1;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"F+F+ " }{TEXT 295 1 " " }{TEXT -1 1 " " }{TEXT 298 3 "and" }{TEXT -1 2 " \+ " }{XPPEDIT 296 0 "x^3+y^3 = 1;" "6#/,&*$%\"xG\"\"$\"\"\"*$%\"yGF'F(F( " }{TEXT -1 2 " " }{TEXT 299 15 "are isomorphic!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "subs(\{x=2^(2/3)/(x+y), y=sqrt(3)*(x-y)/(x+y) \}, y^2-x^3+1):\nfactor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\" \"%\"\"\",(*$)%\"xG\"\"$F&F&*$)%\"yGF+F&F&F&!\"\"F&,&F*F&F.F&!\"$F&" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "subs(\{x=(y+sqrt(3))/2^(1/ 3)/sqrt(3)/x, y=(sqrt(3)-y)/2^(1/3)/sqrt(3)/x\}, x^3+y^3-1):\nfactor(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(*$)%\"yG\"\"#\"\"\"!\"\"F* F+*$)%\"xG\"\"$F*F*F*F.!\"$F+" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 603 3 "If " }{XPPEDIT 18 0 "``(x,y)-`>`;" "6#,&-%!G6$%\"xG%\"yG \"\"\"%\">G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(phi(x,y),psi(x,y) );" "6#-%!G6$-%$phiG6$%\"xG%\"yG-%$psiG6$F)F*" }{TEXT -1 2 " " } {TEXT 604 42 "is an endomorphism of the elliptic curve " }{XPPEDIT 18 0 "x^3+y^3 = 1;" "6#/,&*$%\"xG\"\"$\"\"\"*$%\"yGF'F(F(" }{TEXT 612 1 "," }{TEXT -1 1 "\n" }{TEXT 605 6 "then " }{XPPEDIT 18 0 "z-`>`;" " 6#,&%\"zG\"\"\"%\">G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/(phi(z^(- 1/3))^3);" "6#*&\"\"\"F$*$-%$phiG6#)%\"zG,$*&F$F$\"\"$!\"\"F.F-F." } {TEXT -1 2 " " }{TEXT 606 76 "gives a transformation of the elliptic \+ integral. As in the previous subcase," }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 607 77 "The group of transformations of this hypergeometric e quation into itself is " }{TEXT 608 1 "Z" }{XPPEDIT 18 0 "[omega]^`*` ;" "6#)7#%&omegaG%\"*G" }{TEXT -1 0 "" }{TEXT 610 1 "/" }{TEXT -1 0 " " }{TEXT 609 0 "" }{TEXT -1 0 "" }{XPPEDIT 18 0 "``(1,omega,omega^2,-1 .-omega,-omega^2);" "6#-%!G6'\"\"\"%&omegaG*$F'\"\"#,&-%&FloatG6$F&\" \"!!\"\"F'F/,$*$F'F)F/" }{TEXT -1 0 "" }{TEXT 611 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 262 "" 0 "" {TEXT -1 0 "" }{TEXT 614 18 "Multiplication by " }{TEXT 618 2 "3 " }{TEXT 615 12 "endomorph ism" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "Hyper geom([1/3,2/3],[4/3],z)=(1-z+z^2)*(1-z)^(1/3)/\n(1+3*z-6*z^2+z^3)*Hype rgeom([1/3,2/3],[4/3],\n-27*z*(z-1)*(z^2-z+1)^3/(z^3-6*z^2+3*z+1)^3); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$#\"\"\"\"\"$#\" \"#F*7##\"\"%F*%\"zG**,(F)F)F0!\"\"*$)F0F,F)F)F),&F)F)F0F3F(,*F)F)*&F* F)F0F)F)*&\"\"'F)F5F)F3*$)F0F*F)F)F3-F%6%F'F-,$*,\"#FF)F0F),&F0F)F)F3F )F2F*F7!\"$F3F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "checkser ies(%,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"zG\"\"\"\"\"!#F&\" \"'F&#\"\"&\"#j\"\"##\"\"%\"#\")\"\"$#\"$5\"\"%fJF/#\"#x\"%;HF+-%\"OG6 #F&F)F$" }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT -1 0 "" }{TEXT 616 18 "Mu ltiplication by " }{XPPEDIT 257 0 "3+omega;" "6#,&\"\"$\"\"\"%&omegaGF %" }{TEXT -1 1 " " }{TEXT 617 12 "endomorphism" }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "Hypergeom([1/3,2/3],[4/3],z )=(1-z-z^2/(3*omega+2))/\n(1+(3*omega+2)*z-(3*omega+2)*z^2)*Hypergeom( [1/3,2/3],[4/3],\nz*(z^2+(3*omega+2)*z-3*omega-2)^3/\n(-(3*omega+2)*z^ 2+(3*omega+2)*z+1)^3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*Hypergeo mG6%7$#\"\"\"\"\"$#\"\"#F*7##\"\"%F*%\"zG*(,(F)F)F0!\"\"*&F0F,,&*&F*F) %&omegaGF)F)F,F)F3F3F),(F)F)*&F5F)F0F)F)*&F5F))F0F,F)F3F3-F%6%F'F-*(F0 F),**$F;F)F)F9F)*&F*F)F7F)F3F,F3F*F8!\"$F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "checkseries(%,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#/+1%\"zG\"\"\"\"\"!#F&\"\"'F&#\"\"&\"#j\"\"##\"\"%\"#\")\"\"$#\"$5\" \"%fJF/#\"#x\"%;HF+-%\"OG6#F&F)F$" }}}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 16 "Transformations " }{XPPEDIT 18 0 "` `(1/2,1/3,1/6);" "6#- %\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)*&F'F'\"\"'F)" }{TEXT -1 3 " \+ <" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" }{XPPEDIT 18 0 "` `(1/3,1/3,1/ 3);" "6#-%\"~G6%*&\"\"\"F'\"\"$!\"\"*&F'F'F(F)*&F'F'F(F)" }{TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 619 104 "By considering that these transf ormations naturally correspond to the isogenies from the elliptic curv e " }{XPPEDIT 623 0 "x^3+y^3 = 1;" "6#/,&*$%\"xG\"\"$\"\"\"*$%\"yGF'F( F(" }{TEXT -1 1 " " }{TEXT 622 24 " to the elliptic curve " } {XPPEDIT 620 0 "y^2 = x^3-1;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"F+! \"\"" }{TEXT -1 3 " . " }{TEXT 621 456 "Since these curves are isomorp hic, such isogenies are just their endomorphisms composed with an isom orphism (say, the isomorphism fom the previous section). \nBesides, it does not matter whether we apply \"the same\" isogeny on one or on ot her curve.\nAccordingly, transformations of these functions are (up to fractional-linear transformations) compositions of the following clas sical quadratic transformation and transformations of the previous two subcases." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "hypergeom([1/3,2/3],[4/3],z)=(1-z)^(-1/6)*\nhypergeom([1/2,1/6],[7/6] ,z^2/4/(z-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$# \"\"\"\"\"$#\"\"#F*7##\"\"%F*%\"zG*&,&F)F)F0!\"\"#F3\"\"'-F%6%7$#F)F5# F)F,7##\"\"(F5,$*(F/F3F0F,,&F0F)F)F3F3F)F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "checkseries(%,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#/+1%\"zG\"\"\"\"\"!#F&\"\"'F&#\"\"&\"#j\"\"##\"\"%\"#\")\"\"$#\"$5\" \"%fJF/#\"#x\"%;HF+-%\"OG6#F&F)F$" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 16 "Transformations " }{XPPEDIT 18 0 "` `(1/2,1/3,1/6);" "6#- %\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)*&F'F'\"\"'F)" }{TEXT -1 3 " \+ <" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(2/3,1/6,1/6);" "6#-%\"~G6%*&\"\"#\"\"\"\"\"$!\"\"*&F(F(\"\"'F** &F(F(F,F*" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 624 59 "A hy pergeometric equation with local exponent differences " }{XPPEDIT 18 0 "``(2/3,1/6,1/6);" "6#-%!G6%*&\"\"#\"\"\"\"\"$!\"\"*&F(F(\"\"'F**&F( F(F,F*" }{TEXT -1 1 " " }{TEXT 626 176 "has also some constant or radi cal solutions, other (more interesting) solutions can be represented b y integrals. A representative solution with these local exponent diffe rences" }{TEXT -1 1 " " }{TEXT 625 17 "is the following:" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "z^(1/6)*Hypergeom([1/3,1/6],[7 /6],z)=1/6*\n'int( t^(-5/6)*(1-t)^(-1/3), t=0..z )';\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"zG#\"\"\"\"\"'F(-%*HypergeomG6%7$#F(\"\"$F' 7##\"\"(F)F&F(,$*&F'F(-%$intG6$*&F(F(*&)%\"tG#\"\"&F)F(),&F(F(F;!\"\"# F(F/F(F@/F;;\"\"!F&F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 628 13 "By the change" }{TEXT -1 1 " " }{XPPEDIT 256 0 "t-`>`;" "6#,&%\"tG\"\"\"%\">G!\"\"" }{TEXT 627 1 " " }{XPPEDIT 257 0 "(sqrt(x^ 6+1)-x^3)^2;" "6#*$,&-%%sqrtG6#,&*$%\"xG\"\"'\"\"\"F,F,F,*$F*\"\"$!\" \"\"\"#" }{TEXT -1 2 " " }{TEXT 629 79 "of the integration variable w e get a holomorphic integral on the genus 2 curve " }{XPPEDIT 258 0 "y ^2 = x^6+1;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"'\"\"\"F+F+" }{TEXT 630 2 ". " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "z^(1/6)*Hypergeom([1/2,1/4],[ 5/4],z)=2^(-1/3)*\n'int( x/sqrt(x^6+1), x=(1-z)^(1/3)/2^(1/3)/z^(1/6). .infinity )';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"zG#\"\"\"\"\"' F(-%*HypergeomG6%7$#F(\"\"##F(\"\"%7##\"\"&F1F&F(,$*&F.F(*&)F/#F/\"\"$ F(-%$intG6$*&%\"xGF(-%%sqrtG6#,&*$)F?F)F(F(F(F(!\"\"/F?;,$**F/FF,&F&FF F(F(#F(F:F/F9F&#FFF)F(%)infinityGF(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 631 141 "From comparison of integralrepresentations, it fo lows that transformations between hypergeometric equations with local exponent differences " }{XPPEDIT 18 0 "``(1/2,1/3,1/6);" "6#-%!G6%*& \"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)*&F'F'\"\"'F)" }{TEXT -1 1 " " }{TEXT 632 4 "and " }{XPPEDIT 18 0 "``(2/3,1/6,1/6);" "6#-%!G6%*&\"\"#\"\"\" \"\"$!\"\"*&F(F(\"\"'F**&F(F(F,F*" }{TEXT -1 1 " " }{TEXT 633 52 "corr espond to algebraic maps from the genus 2 curve " }{XPPEDIT 18 0 "y^2 \+ = x^6+1;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"'\"\"\"F+F+" }{TEXT 634 22 " to the genus 1 curve " }{XPPEDIT 18 0 "y^2 = x^3-1;" "6#/*$%\"yG\"\"#,&* $%\"xG\"\"$\"\"\"F+!\"\"" }{TEXT -1 1 " " }{TEXT 635 38 " which map th e two infinite points of " }{XPPEDIT 18 0 "y^2 = x^6+1;" "6#/*$%\"yG\" \"#,&*$%\"xG\"\"'\"\"\"F+F+" }{TEXT -1 1 " " }{TEXT 636 26 "to the inf inite point of " }{XPPEDIT 18 0 "y^2 = x^3-1;" "6#/*$%\"yG\"\"#,&*$% \"xG\"\"$\"\"\"F+!\"\"" }{TEXT -1 0 "" }{TEXT 637 61 ". These algebrai c maps are compositions of the quadratic map " }{XPPEDIT 18 0 "``(x,y) ;" "6#-%!G6$%\"xG%\"yG" }{XPPEDIT 18 0 "-`>`;" "6#,$%\">G!\"\"" } {XPPEDIT 18 0 "``(-x^2,i*y);" "6#-%!G6$,$*$%\"xG\"\"#!\"\"*&%\"iG\"\" \"%\"yGF-" }{TEXT -1 1 " " }{TEXT 638 25 "and an endomorphisms of " } {XPPEDIT 18 0 "y^2 = x^3-1;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"F+!\" \"" }{TEXT -1 0 "" }{TEXT 639 221 ". Correspondingly, the hypergeometr ic transformations factor through the following classical quadratic tr ansformation, and previously considered transformations of hypergeomet ric functions with local exponent differences " }{XPPEDIT 18 0 "``(1/2 ,1/3,1/6);" "6#-%!G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)*&F'F'\"\"'F)" } {TEXT 640 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "hypergeom( [1/3,1/6],[7/6],z)=(1-z)^(-1/3)*\nhypergeom([1/2,1/6],[7/6],-4*z/(z-1) ^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$#\"\"\"\"\"' #F)\"\"$7##\"\"(F*%\"zG*&,&F0!\"\"F)F)#F3F,-F%6%7$F(#F)\"\"#F-,$*(\"\" %F)F0F),&F0F)F)F3!\"#F3F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "checkseries(%,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"zG\"\" \"\"\"!#F&\"#@F&#\"\"#\"$<\"F+#\"#9\"%R:\"\"$#\"\"(\"%:7\"\"%#\"#\"*\" &*fA\"\"&-%\"OG6#F&\"\"'F$" }}}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 35 "Hiperbolic hypergeometric functions" }}{PARA 259 "" 0 "" {TEXT -1 93 "Here we consider transformations of hypergeometric functions with \+ local exponent differences " }{XPPEDIT 641 0 "``(1/k,1/m,1/n);" "6#-%! G6%*&\"\"\"F'%\"kG!\"\"*&F'F'%\"mGF)*&F'F'%\"nGF)" }{TEXT -1 8 ", wher e " }{XPPEDIT 642 0 "k,m,n;" "6%%\"kG%\"mG%\"nG" }{TEXT -1 18 " are in tegers and " }{XPPEDIT 643 0 "1/k+1/m+1/n < 1;" "6#2,(*&\"\"\"F&%\"kG! \"\"F&*&F&F&%\"mGF(F&*&F&F&%\"nGF(F&F&" }{TEXT -1 50 ". We have a fin ite list of these transformations." }}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 24 "Degree 8 transformation " }{XPPEDIT 18 0 "` `(1/2,1/3,1/7);" "6 #-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)*&F'F'\"\"(F)" }{TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/3,1/3,1/7);" "6#-%\"~G6%*&\"\"\"F'\"\"$!\"\"*&F'F'F(F)*&F' F'\"\"(F)" }{TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 12 "Recall that " }{XPPEDIT 256 0 "omega;" "6#%&omegaG" }{TEXT -1 14 " is a root of \+ " }{XPPEDIT 257 0 "omega^2+omega+1 = 0;" "6#/,(*$%&omegaG\"\"#\"\"\"F& F(F(F(\"\"!" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "sbs1:= psi[1]=1/64*x*(x-1)/(6*x*omega+3*x-8-3*omega)^7*\n(27*x^2- 723*x-1392*x*omega-496+696*omega)^3: sbs1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$psiG6#\"\"\",$*,\"#k!\"\"%\"xGF',&F,F'F'F+F',**(\" \"'F'F,F'%&omegaGF'F'*&\"\"$F'F,F'F'\"\")F+*&F3F'F1F'F+!\"(,,*&\"#FF') F,\"\"#F'F'*&\"$B(F'F,F'F+*(\"%#R\"F'F,F'F1F'F+\"$'\\F+*&\"$'pF'F1F'F' F3F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "Hyp1:= Hypergeom([ 2/21,5/21],[2/3],x)=\n(1-(39*omega+33)/49*x)^(-1/12)*Hypergeom([1/84,1 3/84],[2/3],psi[1]);\ncheckseries( subs(sbs1,%), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Hyp1G/-%*HypergeomG6%7$#\"\"#\"#@#\"\"&F,7##F+\" \"$%\"xG*&,&\"\"\"F5*(\"#\\!\"\",&*&\"#RF5%&omegaGF5F5\"#LF5F5F2F5F8#F 8\"#7-F'6%7$#F5\"#%)#\"#8FDF/&%$psiG6#F5F5" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"xG\"\"\"\"\"!#\"\"&\"$Z\"F&#\"$*H\"&4;#\"\"##\"'$ ea\"\")Q\"f!>\"\"$#\")l&Gb\"\"+'G$p,G\"\"%#\",^Oc&)=\"\".%H\"RUH)GF)-% \"OG6#F&\"\"'F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 682 32 "Use an 'otherhpgid' arrangement:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Hyp1a:= otherhpgid( Hyp1, \"CbA\", \"CbA\", sbs1 );\ncheckseries ( %, x );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Hyp1aG/-%*HypergeomG6% 7$#\"\"#\"#@#\"\"$\"\"(7##\"\"'F/%\"xG*(,&\"\"\"F6F3!\"\"#F7\"#%),(** \"\")F6\"#FF7,&\"#iF7*&\"#()F6%&omegaGF6F6F6F3F+F6*(\"\"*F7,&\"$T#F6*& \"$k%F6FBF6F6F6F3F6F7F6F6#F7\"#G-F'6%7$#F6F9#\"#HF9F0,$*.\"#kF6,&*&F.F 6FBF6F6F+F6F6F3F6,(*&\"#\\F6F3F6F6*&\"#RF6FBF6F7\"#LF7F/,&F3F6F6F7F7,, \"%BSF7*&\"&xL\"F6F3F6F7*(\"'0#\\\"F6F3F6FBF6F6*&\"'cW8F6)F3F+F6F6*&\" %\\BF6FBF6F7!\"$F6F6" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"xG\"\"\" \"\"!#F&\"#@F&#\"$:\"\"%Ld\"\"##\"%,V\"'z6O\"\"$#\"&5I%\"(()4D&\"\"%# \"')\\u'\"*F2F5\"\"\"&-%\"OG6#F&\"\"'F$" }}}}{SECT 1 {PARA 260 "" 0 " " {TEXT -1 24 "Degree 9 transformation " }{XPPEDIT 18 0 "` `(1/2,1/3,1 /7);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)*&F'F'\"\"(F)" } {TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/2,1/7,1/7);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F' F'\"\"(F)*&F'F'F+F)" }{TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 12 "R ecall that " }{XPPEDIT 256 0 "xi;" "6#%#xiG" }{TEXT -1 14 " is a root \+ of " }{XPPEDIT 257 0 "xi^2+xi+2 = 0;" "6#/,(*$%#xiG\"\"#\"\"\"F&F(F'F( \"\"!" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "sb s2:= psi[2]=27/49*x*(x-1)*(49*x-31-13*xi)^7/\n(7203*x^3+(9947*xi-5831) *x^2-(9947*xi+2009)*x-87*xi+275)^3: sbs2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$psiG6#\"\"#,$*.\"#F\"\"\"\"#\\!\"\"%\"xGF+,&F.F+F+F -F+,(*&F,F+F.F+F+\"#JF-*&\"#8F+%#xiGF+F-\"\"(,,*&\"%.sF+)F.\"\"$F+F+*& ,&*&\"%Z**F+F5F+F+\"%JeF-F+)F.F'F+F+*&,&*&F?F+F5F+F+\"%4?F+F+F.F+F-*& \"#()F+F5F+F-\"$v#F+!\"$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "Hyp2:= Hypergeom([3/28,17/28],[6/7],x)=\n(1-7/8*(-10+29*xi)*x+343 /512*(29*xi-50)*x^2+1029/16384*(87*xi+362)*x^3)^(-1/28)*Hypergeom([1/8 4,29/84],[6/7],psi[2]);\ncheckseries( subs(sbs2,%), x);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%%Hyp2G/-%*HypergeomG6%7$#\"\"$\"#G#\"#F6F6\"#]F9F6F3\"\"#F6**\"%H5F6\"&%Q;F9,& *&\"#()F6F>F6F6\"$i$F6F6F3F+F6#F9F,-F'6%7$#F6\"#%)#F=FRF/&%$psiG6#FEF6 " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"xG\"\"\"\"\"!#\"#<\"$C#F&#\" &:P#\"')G_'\"\"##\"(n$4o\"*C]A#H\"\"$#\"+\")>@[m\"-cAV]FR\"\"%#\"-x(3& pnQ\"/[%3cOD$H\"\"&-%\"OG6#F&\"\"'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Hyp2a:= otherhpgid( Hyp2, \"Cba\", \"bac\", sbs2 );\n checkseries( %, x );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Hyp2aG/-%*H ypergeomG6%7$#\"\"\"\"\"%#\"\"$\"#G7##F+\"\"#%\"xG*&,**(\"%.s!\"\",&\" $v#F8*&\"#()F+%#xiGF+F+F+F3F.F8*(\"$Z\"F8,&\"#TF+*&\"$.#F+F=F+F+F+F3F2 F8*(\"#@F8,&\"# " 0 "" {MPLTEXT 1 0 77 "sbs3:= psi[3]=-x^2*(x-1)*(-8 1+49*x)^7/4/\n(6561-13851*x-9261*x^2+16807*x^3)^3;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%sbs3G/&%$psiG6#\"\"$,$*,\"\"%!\"\"%\"xG\"\"#,&F.\" \"\"F1F-F1,&\"#\")F-*&\"#\\F1F.F1F1\"\"(,*\"%hlF1*&\"&^Q\"F1F.F1F-*&\" %h#*F1)F.F/F1F-*&\"&2o\"F1)F.F)F1F1!\"$F-" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 161 "Hyp3:= Hypergeom( [5/42,19/42], [5/7], x)=\n(1-19/ 9*x-343/243*x^2+16807/6561*x^3)^(-1/28)*\nHypergeom( [1/84,29/84], [6/ 7], psi[3]);\ncheckseries( subs(sbs3,%), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Hyp3G/-%*HypergeomG6%7$#\"\"&\"#U#\"#>F,7##F+\"\"(% \"xG*&,*\"\"\"F5*&#F.\"\"*F5F2F5!\"\"*&#\"$V$\"$V#F5*$)F2\"\"#F5F5F9*& #\"&2o\"\"%hlF5*$)F2\"\"$F5F5F5#F9\"#G-F'6%7$#F5\"#%)#\"#HFN7##\"\"'F1 &%$psiG6#FGF5" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"xG\"\"\"\"\"!# \"#>\"$_#F&#\"&tW&\"('4C:\"\"##\")*y\"GE\"+wl@_6\"\"$#\"-b?eA#*\\\"/3Q -#H(>I\"\"%#\"0^ZwAFk$H\"2[))**pF:HG#\"\"&-%\"OG6#F&\"\"'F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Hyp3a:= otherhpgid( Hyp3, \" bac\", \"abc\", sbs3 );\ncheckseries( %, x );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Hyp3aG/-%*HypergeomG6%7$#\"\"&\"#U#\"#>F,7##\"\"'\" \"(%\"xG*&,*\"\"\"F6*&#\"$T\"\"\"#F6F3F6!\"\"*&#\"%X^\"#KF6*$)F3F:F6F6 F6*&#\"&2o\"\"$c#F6*$)F3\"\"$F6F6F;#F;\"#G-F'6%7$#F6\"#%)#\"#HFOF/,$*, \"\"%F;,&F3F6F6F;F:F3F6,&F?F6*&\"#\\F6F3F6F6F2,*FEF;*&\"&[!=F6F3F6F6*& \"&g6%F6FAF6F;*&FDF6FGF6F6!\"$F6F6" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# /+1%\"xG\"\"\"\"\"!#\"#&*\"%7:F&#\"'lBF\"(Cm!**\"\"##\"*\"*RN*\\\",w4j d*H\"\"$#\".X!f!H_[*\"0;Gk%)oK:)\"\"%#\"1FD()z\"y+h$\"3k#z*f)\\Z#4T\" \"&-%\"OG6#F&\"\"'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Hyp 3b:= otherhpgid( Hyp3, \"Cba\", \"CbA\", sbs3 );\ncheckseries( %, x ); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Hyp3bG/-%*HypergeomG6%7$#\"#<\" #U#\"\"&F,7##\"\"#\"\"$%\"xG*(,&\"\"\"F6F3!\"\"#F7\"#%),&*(\"#\")F6\"# \\F7F3F6F7F6F6#F7\"#7-F'6%7$#F6F9#\"#8F9F/,$*,\"\"%F6F3F6,**&\"%hlF6)F 3F2F6F6*&\"&^Q\"F6)F3F1F6F7*&\"%h#*F6F3F6F7\"&2o\"F6F2,&F3F6F6F7F7,&*& F\"\"$#\"-ZY ,Z;s\"/CyzUH!f%\"\"%#\"1ZavSDE>Y\"3oro6^-tyP\"\"&-%\"OG6#F&\"\"'F$" }} }}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 25 "Degree 18 transformation " } {XPPEDIT 18 0 "` `(1/2,1/3,1/7);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F '\"\"$F)*&F'F'\"\"(F)" }{TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%! G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/7,1/7,2/7);" "6#-%\"~G6% *&\"\"\"F'\"\"(!\"\"*&F'F'F(F)*&\"\"#F'F(F)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 684 132 "This is a composition of transfo rmations of degree 9 and 2. The intermediate hypergeometric equation h as local exponent differences " }{XPPEDIT 18 0 "``(1/2,1/7,1/7);" "6#- %!G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"(F)*&F'F'F+F)" }{TEXT -1 0 "" } {TEXT 685 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Hyp18:= hp gcomposition( Quadr2, Hyp2, a=3/14, b=1/2, sbs2 );\ncheckseries(%, x); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Hyp18G/-%*HypergeomG6%7$#\"\"$ \"#9#\"\"&F,7##F.\"\"(%\"xG*(\"\"#F0,:*&\"&el\"\"\"\")F2\"\"'F8!\"\"*( \"%F8F8*(\"(SYq) F8)F2\"\"%F8F>F8F;*&\"(![GAF8FFF8F8*&\"(g*3sF8)F2F+F8F;*(\")3QI8F8FLF8 F>F8F8*(\"(/>l'F8)F2F4F8F>F8F;*&\"(?fA'F8FQF8F8*&\"(Gd9$F8F2F8F;\"(w&[ 5F8#F;\"#G-F'6%7$#F8\"#%)#\"#HFgn7##F:F1,$*0\"#FF8\"$7&F;,&*&F1F8F>F8F 8\"#5F8F8,&F2F8F8F;F8F2F4,,*&\"#KF8FQF8F8*&\"#*)F8F2F8F8FhoF;*(\"#\"*F 8F2F8F>F8F8*&FjoF8F>F8F;F1,<\"%T')F;*&\"&Bf#F8F2F8F8*&\"$c#F8F9F8F8*( \"'Z?:F8FQF8F>F8F8*(\"'4XEF8FLF8F>F8F;*(\"&%)f#F8FAF8F>F8F;*(\"'!Ga\"F 8FFF8F>F8F8*&\"&T\"\\F8FQF8F8*&\"'!38\"F8FFF8F8*&\"'([T\"F8FLF8F;*&\"& ;!QF8FAF8F;*&F=F8F>F8F8*(\"&^P#F8F2F8F>F8F;!\"$F;F8" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/+1%\"xG\"\"\"\"\"!#\"\"$\"#GF&#\"$B$\"%si\"\"##\"%(z &\"';cV\"\"\"&-%\"OG6#F&\"\" 'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Hyp18a:= otherhpgid( Hyp18, \"bac\", \"abc\" );\ncheckseries(%, x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Hyp18aG/-%*HypergeomG6%7$#\"\"$\"#9#\"\"&F,7##\"\"' \"\"(%\"xG*&,0\"\"\"F6*(\"\"#!\"\",&\"$&GF6*&\"$.#F6%#xiGF6F6F6F3F6F6* $)F3F1F6F6**F.F6\"#KF9,&\"%H=F9*&\"$4'F6F>F6F6F6F3F8F6**F.F6FBF9FCF6F3 \"\"%F6*(\"$c#F9,&\"&2U'F9*&\"&rF*F6F>F6F6F6F3F+F9*(F8F9F:F6F3F.F6#F9 \"#G-F'6%7$#F6\"#%)#\"#HFVF/,$*0\"#FF6\"$7&F9,&*&F2F6F>F6F6\"#5F6F6F3F 6,&F3F6F6F9F8,*FBF6*&\"$`\"F6F3F6F9*&FBF6)F3F8F6F6*(\"#\"*F6F3F6F>F6F9 F2,:*&\"&![OF6F3F6F6*&FJF6F@F6F6*&\"&gJ(F6F_oF6F9*&FgoF6)F3FHF6F9*&FLF 6)F3F+F6F6*&FdoF6)F3F.F6F6*(\"&gV#F6F_oF6F>F6F6*(FNF6F[pF6F>F6F9*(F_pF 6FioF6F>F6F6*(\"&%)f#F6F]pF6F>F6F6*(FcpF6F3F6F>F6F6FJF6!\"$F6F6" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"xG\"\"\"\"\"!#\"\"&\"#cF&#\"%:; \"&o2%\"\"##\"'V,6\"(;gc%\"\"$#\")0O)e#\"+w8=M:\"\"%#\"+(GUf4\"\",cq:9 f)F)-%\"OG6#F&\"\"'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 25 "Degree 24 transformation " } {XPPEDIT 18 0 "` `(1/2,1/3,1/7);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F '\"\"$F)*&F'F'\"\"(F)" }{TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%! G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/7,1/7,1/7);" "6#-%\"~G6% *&\"\"\"F'\"\"(!\"\"*&F'F'F(F)*&F'F'F(F)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 686 132 "This is a composition of transforma tions of degree 8 and 3. The intermediate hypergeometric equation has \+ local exponent differences " }{XPPEDIT 18 0 "``(1/3,1/3,1/7);" "6#-%!G 6%*&\"\"\"F'\"\"$!\"\"*&F'F'F(F)*&F'F'\"\"(F)" }{TEXT -1 0 "" }{TEXT 687 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Hyp24:= hpgcomposition( Cubic4, Hyp1a, a=2/7, sbs1 ); \ncheckseries(%, x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Hyp24G/-%*H ypergeomG6%7$#\"\"#\"\"(#\"\"$F,7##\"\"'F,%\"xG**,(\"\"\"F5F2!\"\"*&F2 F5%&omegaGF5F6#F6\"#G,&F7F5F5F5F9,0F5F5*&\"$N#F5F2F5F6*$)F2F1F5F5*&\"% I9F5)F2F+F5F5*&\"$q#F5)F2\"\"%F5F5*&\"%&p\"F5)F2F.F5F6*&\"$H#F5)F2\"\" &F5F5F9-F'6%7$#F5\"#%)#\"#HFSF/,$*0\"%G " 0 "" {MPLTEXT 1 0 165 "1728*x*(x-1)*(1+5*x-8*x^2+x^3)^7/(1-235*x+x^6+1430*x ^2+270*x^4-1695*x^3+229*x^5)^3/(x^2-x+1)^3;\nevala(Normal( % / op(3, s elect(Q->op(0,Q)=Hypergeom,op(2,Hyp24)) )));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.\"%G<\"\"\"%\"xGF&,&F'F&F&!\"\"F&,*F&F&*&\"\"&F&F'F &F&*&\"\")F&)F'\"\"#F&F)*$)F'\"\"$F&F&\"\"(,0F&F&*&\"$N#F&F'F&F)*$)F' \"\"'F&F&*&\"%I9F&F/F&F&*&\"$q#F&)F'\"\"%F&F&*&\"%&p\"F&F2F&F)*&\"$H#F &)F'F,F&F&!\"$,(*$F/F&F&F'F)F&F&FFF&" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 25 "Degree 10 transformation " }{XPPEDIT 18 0 "` `(1/2,1/3,1/8);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)* &F'F'\"\")F)" }{TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/3, 1/8, 1/8);" "6#-%\"~G6%*&\"\" \"F'\"\"$!\"\"*&F'F'\"\")F)*&F'F'F+F)" }{TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 12 "Recall that " }{XPPEDIT 256 0 "beta;" "6#%%betaG" } {TEXT -1 14 " is a root of " }{XPPEDIT 257 0 "beta^2+2 = 0;" "6#/,&*$% %betaG\"\"#\"\"\"F'F(\"\"!" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 118 "sbs4:= psi[4]=4*x*(x-1)*(8*beta*x-4*beta+7)^8/\n(2 048*beta*x^3-3072*beta*x^2-3264*x^2+912*beta*x+3264*x\n+56*beta-17)^3; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sbs4G/&%$psiG6#\"\"%,$*,F)\"\" \"%\"xGF,,&F-F,F,!\"\"F,,(*(\"\")F,%%betaGF,F-F,F,*&F)F,F3F,F/\"\"(F,F 2,0*(\"%[?F,F3F,)F-\"\"$F,F,*(\"%sIF,F3F,)F-\"\"#F,F/*&\"%kKF,F=F,F/*( \"$7*F,F3F,F-F,F,*&F@F,F-F,F,*&\"#cF,F3F,F,\"# " 0 "" {MPLTEXT 1 0 198 "Hyp4:= Hypergeom([5/24,13/24],[7/8 ],x)=\n(1+16*(4-17*beta)/9*x-64*(167-136*beta)/243*x^2\n+2048*(112-17* beta)/6561*x^3)^(-1/16)*\nHypergeom([1/48,17/48],[7/8], psi[4] );\nche ckseries( subs(sbs4,%), x);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%Hy p4G/-%*HypergeomG6%7$#\"\"&\"#C#\"#8F,7##\"\"(\"\")%\"xG*&,*\"\"\"F6** \"#;F6\"\"*!\"\",&\"\"%F6*&\"#F6F?F 6F:F6F3\"\"$F6#F:F8-F'6%7$#F6\"#[#F>FTF/&%$psiG6#F " 0 "" {MPLTEXT 1 0 68 "Hyp4a:= otherhpgid( Hyp4, \"CbA\", \"CbA\", sbs4 );\ncheckseries( \+ %, x );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Hyp4aG/-%*HypergeomG6%7$ #\"\"&\"#C#\"\"\"\"\"$7##\"\"#F/%\"xG*(,&F.F.F3!\"\"#F6\"#[,&F.F.*(\"# ;F6,&\"\")F.*&\"\"(F.%%betaGF.F.F.F3F.F6#F6\"\"'-F'6%7$#F.F8#F?F8F0,$* .\"\"%F6,&\"#F.F@F.FWF.F6*&\"%hlF.)F3F/F. F.*&\"&;[$F.F@F.F6F/,&F3F.F.F6F6,(\"#kF6*&FNF.F@F.F.*&\"#\")F.F3F.F.! \")F.F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"xG\"\"\"\"\"!#\"\"&\" #[F&#\"#H\"$w&\"\"##\"&f2\"\"'wTk(F)-%\"OG6#F&\"\"'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 25 "Degree 12 transformation \+ " }{XPPEDIT 18 0 "` `(1/2,1/3,1/8);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*& F'F'\"\"$F)*&F'F'\"\")F)" }{TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#, $%!G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/4,1/8,1/8);" "6#-%\"~ G6%*&\"\"\"F'\"\"%!\"\"*&F'F'\"\")F)*&F'F'F+F)" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 688 137 "This is a composition of transformations of degree 3, 2 and 2. The intermediate hypergeometric equations have local exponent differences " }{XPPEDIT 18 0 "``(1/2,1/ 4,1/8);" "6#-%!G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"%F)*&F'F'\"\")F)" } {TEXT 689 6 " and " }{XPPEDIT 18 0 "``(1/2,1/8,1/8);" "6#-%!G6%*&\"\" \"F'\"\"#!\"\"*&F'F'\"\")F)*&F'F'F+F)" }{TEXT 692 1 "." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 693 249 "Recall that 'Sextic2' is already a composition of two quadratic transformations. \nOf course, \+ the degree 12 transformation can be expressed as composition of two tr ansformations\n(of degree 3 and 4) via the other intermediate hypergeo metric equation. " }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Hyp12:= hpgcomposition( subs(a=1/4,b=1/2,Quadr2),\nsubs(a=1/16,Sextic 2) );\ncheckseries(%, x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Hyp12G /-%*HypergeomG6%7$#\"\"\"\"\"%#\"\"$\"\")7##F.F,%\"xG*(\"\"#F*,,*$)F2F ,F+F+*&\"#cF+)F2F.F+F+*&\"#SF+)F2F4F+!\"\"*&\"#KF+F2F+F>\"#;F+#F>FA-F' 6%7$#F+\"#[#\"#F/F2F4,&F2F+F+F>F+F5! \"$F+F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"xG\"\"\"\"\"!#F&\"\") F&#\"#b\"$'*)\"\"##\"$&G\"%or\"\"$#\"%pm\"'w$H#\"\"%#\"%nf\"'W@E\"\"&- %\"OG6#F&\"\"'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Hyp12a: = otherhpgid( Hyp12, \"Cba\", \"abc\" );\ncheckseries( %, x );" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Hyp12aG/-%*HypergeomG6%7$#\"\"\"\" \"##F+\"\"%7##\"\"(\"\")%\"xG*&,,F+F+*&\"#cF+F3F+F+*&\"#SF+)F3F,F+!\" \"*&\"#KF+)F3\"\"$F+F;*&\"#;F+)F3F.F+F+#F;FA-F'6%7$#F+\"#[#\"#\"%/d\"\"%#\"$`\"F3\"\"&-%\"OG6#F&\"\"'F$" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 25 "Degree 12 transformation " }{XPPEDIT 18 0 "` `(1/2,1/3,1/9);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)* &F'F'\"\"*F)" }{TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/9, 1/9, 1/9);" "6#-%\"~G6%*&\"\" \"F'\"\"*!\"\"*&F'F'F(F)*&F'F'F(F)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 690 132 "This is a composition of transformations of degree 4 and 3. The intermediate hypergeometric equation has local exponent differences " }{XPPEDIT 18 0 "``(1/3,1/3,1/9);" "6#-%!G6%*& \"\"\"F'\"\"$!\"\"*&F'F'F(F)*&F'F'\"\"*F)" }{TEXT -1 0 "" }{TEXT 691 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Hyp12b:= hpgcomposit ion( subs(a=1/3,Cubic4), subs(a=1/12,Quartic1) );\ncheckseries(%, x); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Hyp12bG/-%*HypergeomG6%7$#\"\" \"\"\"$#\"\"%\"\"*7##\"\")F/%\"xG*(,(F+F+F3!\"\"*&F3F+%&omegaGF+F6#F6 \"#7,.F+F+*(\"#^F+F3F+F8F+F6*&\"#FF+F3F+F6*(F=F+F8F+)F3\"\"#F+F+*&\"#C F+FAF+F+*$)F3F,F+F+F9-F'6%7$#F+\"#O#\"#8FKF0,$*0\"$#>F+,&*&FBF+F8F+F+F +F+F+F3F+,&F3F+F+F6F+,(F3F+F+F6F8F6F/,&F3F+F8F+!\"$F;FVF+F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"xG\"\"\"\"\"!#F&\"\"'F&#\"#8\"$`\"\"\" ##\"#x\"%x8\"\"$#\"$T$\"%i#)\"\"%#\"$.%\"&$R7\"\"&-%\"OG6#F&F)F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 24 "Degree 6 transformation " }{XPPEDIT 18 0 "` `(1/2,1/4, 1/5);" "6#-%\"~G6%*&\"\"\"F'\"\"#!\"\"*&F'F'\"\"%F)*&F'F'\"\"&F)" } {TEXT -1 3 " <" }{XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` `(1/4,1/4,1/5);" "6#-%\"~G6%*&\"\"\"F'\"\"%!\"\"*&F' F'F(F)*&F'F'\"\"&F)" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sbs5:= psi[5]=4*I*x*(x-1)*(4*x-2-11*I)^4/(8*x-4+3*I)^ 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sbs5G/&%$psiG6#\"\"&*,^#\"\"% \"\"\"%\"xGF-,&F.F-F-!\"\"F-,&*&F,F-F.F-F-^$!\"#!#6F-F,,&*&\"\")F-F.F- F-^$!\"%\"\"$F-!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "Hyp 5:= Hypergeom( [3/20,7/20], [3/4], x )= (1-8*(4+3*I)/25*x)^(-1/8)*Hype rgeom( [1/40,9/40], [3/4], psi[5] );\ncheckseries( subs(sbs5,%), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Hyp5G/-%*HypergeomG6%7$#\"\"$\"#? #\"\"(F,7##F+\"\"%%\"xG*&,&\"\"\"F5*&^$#!#K\"#D#!#CF:F5F2F5F5#!\"\"\" \")-F'6%7$#F5\"#S#\"\"*FDF/&%$psiG6#\"\"&F5" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"xG\"\"\"\"\"!#\"\"(\"$+\"F&#\"$@'\"&++#\"\"##\"'Z $=%\")+++A\"\"$#\"*HUh)e\",++++S%\"\"%#\".4wM$Q]U\"0++++++=%\"\"&-%\"O G6#F&\"\"'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Hyp5a:= oth erhpgid( Hyp5, \"CbA\", \"CbA\", sbs5 );\ncheckseries( %, x );" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Hyp5aG/-%*HypergeomG6%7$#\"\"$\"#?# \"\"#\"\"&7##\"\"%F/%\"xG*(,&\"\"\"F6F3!\"\"#F7\"#S,&F6F6*&^$#F7F.#!#6 F2F6F3F6F6#F7\"#5-F'6%7$#F6F9#\"#6F9F0*,^$#!\"$F2F6F6F3F6,&*&\"#DF6F3F 6F6^$!#K!#CF6F/,&F3F6F6F7F7,&^$!\")\"#WF6*&\"$D\"F6F3F6F6!\"%F6" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/+1%\"xG\"\"\"\"\"!#\"\"$\"#SF&#\"$h\" \"%+[\"\"##\"$*)*\"&+![F)#\"'tIN\")++KC\"\"%#\"*$)=X2\"\"++++G(*\"\"&- %\"OG6#F&\"\"'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 260 "" 0 "" {TEXT -1 30 "Computaton of finite coverings " }}{PARA 259 "" 0 "" {TEXT -1 59 "Here we illustrate a method for com puting finite coverings " }{TEXT 694 1 "P" }{XPPEDIT 18 0 "``^`1`-`>`; " "6#,&)%!G%\"1G\"\"\"%\">G!\"\"" }{TEXT -1 1 " " }{TEXT 695 1 "P" } {XPPEDIT 18 0 "``^`1`;" "6#)%!G%\"1G" }{TEXT -1 105 " ramified over th ree points with prescribed ramification pattern. See also Algorithm 3. 1 in the mentione " }{TEXT 696 22 "arXiv:math.CA/0310436 " }{TEXT 697 6 "paper." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 698 32 "We choose th e 12 transformation " }{XPPEDIT 18 0 "` `(1/2,1/3,1/8);" "6#-%\"~G6%*& \"\"\"F'\"\"#!\"\"*&F'F'\"\"$F)*&F'F'\"\")F)" }{TEXT -1 3 " <" } {XPPEDIT 18 0 "-``;" "6#,$%!G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` \+ `(1/4,1/8,1/8);" "6#-%\"~G6%*&\"\"\"F'\"\"%!\"\"*&F'F'\"\")F)*&F'F'F+F )" }{TEXT -1 0 "" }{TEXT 699 24 ". We choose the points " }{XPPEDIT 18 0 "z = 0,x = 0,x = 1;" "6%/%\"zG\"\"!/%\"xGF%/F'\"\"\"" }{TEXT -1 1 " " }{TEXT 700 34 "to have local exponent difference " }{XPPEDIT 18 0 "1/8;" "6#*&\"\"\"F$\"\")!\"\"" }{TEXT -1 0 "" }{TEXT 701 43 ", and \+ we assign local exponent differences " }{XPPEDIT 18 0 "1/2,1/3,1/4;" " 6%*&\"\"\"F$\"\"#!\"\"*&F$F$\"\"$F&*&F$F$\"\"%F&" }{TEXT 702 15 " to t he points " }{XPPEDIT 18 0 "z = 1,z = infinity,x = infinity;" "6%/%\"z G\"\"\"/F$%)infinityG/%\"xGF'" }{TEXT -1 2 " \n" }{TEXT 703 13 "respec tively." }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 28 "The target morphism w ill be " }{XPPEDIT 704 0 "z = phi(x);" "6#/%\"zG-%$phiG6#%\"xG" } {TEXT -1 7 ". Here " }{XPPEDIT 705 0 "alpha,a,b,c,d,p;" "6(%&alphaG%\" aG%\"bG%\"cG%\"dG%\"pG" }{TEXT -1 33 " are constants to be determined ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "F0:= x*(x-1)*(x+p)^8;\nFi:= (x ^4+a*x^3+b*x^2+c*x+d)^3;\nphi:= alpha*F0/Fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#F0G*(%\"xG\"\"\",&F&F'F'!\"\"F'),&F&F'%\"pGF'\"\")F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FiG*$),,*$)%\"xG\"\"%\"\"\"F,*& %\"aGF,)F*\"\"$F,F,*&%\"bGF,)F*\"\"#F,F,*&%\"cGF,F*F,F,%\"dGF,F0F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG*,%&alphaG\"\"\"%\"xGF',&F(F'F' !\"\"F',&F(F'%\"pGF'\"\"),,*$)F(\"\"%F'F'*&%\"aGF')F(\"\"$F'F'*&%\"bGF ')F(\"\"#F'F'*&%\"cGF'F(F'F'%\"dGF'!\"$" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 39 "Consider the logarithmic derivative of " }{XPPEDIT 706 0 "phi(x);" "6#-%$phiG6#%\"xG" }{TEXT -1 52 ". The roots of the denomina tor are the points above " }{XPPEDIT 707 0 "z = 0;" "6#/%\"zG\"\"!" } {TEXT -1 5 " and " }{XPPEDIT 256 0 "z = infinity;" "6#/%\"zG%)infinity G" }{TEXT -1 18 "with multiplicity " }{TEXT 708 1 "1" }{TEXT -1 100 ". The roots of the numerator are the remaining ramification points with mutiplicities diminished by " }{TEXT 709 1 "1" }{TEXT -1 64 ". Since \+ we aim to have the remaining ramification points above " }{XPPEDIT 256 0 "z = 1;" "6#/%\"zG\"\"\"" }{TEXT -1 15 " with indices " }{TEXT 710 1 "2" }{TEXT -1 75 ", the numerator of the logaritmic derivative i s presicely the numerator of " }{XPPEDIT 711 0 "1-phi(x);" "6#,&\"\"\" F$-%$phiG6#%\"xG!\"\"" }{TEXT -1 28 ", up to a constant multiple." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "ldphi:= factor( diff(phi,x)/phi ); \nF1:= collect( numer(ldphi), x )^2;\nphi1:= F0/F1;" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%&ldphiG*,,H**\"\"(\"\"\")%\"xG\"\"%F)%\"pGF)%\"aGF) !\"\"**\"\"#F)F+F)F-F)%\"dGF)F)*()F+F1F)F-F)%\"cGF)F/**F1F)F-F)F5F)F+F )F)**F,F))F+\"\"$F)F-F)%\"bGF)F/**\"\"&F)F-F)F:F)F4F)F)**\"\")F)F-F)F. F)F8F)F)*&F-F)F2F)F/*(\"#6F)F-F)F*F)F)*(\"\"*F)F+F)F2F)F/*(\"\"'F)F5F) F4F)F/*(F9F)F:F)F8F)F/*(\"#5F))F+F%#F1G*$),0*&\"\"#\"\"\")%\"xG\" \"'F*!\"\"*&,(*&\"#5F*%\"pGF*F.%\"aGF*\"\"$F*F*)F,\"\"&F*F**&,(*&\"#6F *F3F*F**(\"\"(F*F3F*F4F*F.*&\"\"%F*%\"bGF*F*F*)F,F?F*F**&,**(F?F*F3F*F @F*F.*(\"\")F*F3F*F4F*F**&F5F*F@F*F.*&F=F*%\"cGF*F*F*)F,F5F*F**&,**&F2 F*%\"dGF*F**&F3F*FIF*F.*&F-F*FIF*F.*(F7F*F3F*F@F*F*F*)F,F)F*F**&,(*(F) F*F3F*FNF*F**&\"\"*F*FNF*F.*(F)F*F3F*FIF*F*F*F,F*F**&F3F*FNF*F.F)F*" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%phi1G**%\"xG\"\"\",&F&F'F'!\"\"F', &F&F'%\"pGF'\"\"),0*&\"\"#F')F&\"\"'F'F)*&,(*&\"#5F'F+F'F)%\"aGF'\"\"$ F'F')F&\"\"&F'F'*&,(*&\"#6F'F+F'F'*(\"\"(F'F+F'F6F'F)*&\"\"%F'%\"bGF'F 'F')F&FAF'F'*&,**(FAF'F+F'FBF'F)*(F,F'F+F'F6F'F'*&F7F'FBF'F)*&F?F'%\"c GF'F'F')F&F7F'F'*&,**&F5F'%\"dGF'F'*&F+F'FJF'F)*&F1F'FJF'F)*(F9F'F+F'F BF'F'F')F&F/F'F'*&,(*(F/F'F+F'FOF'F'*&\"\"*F'FOF'F)*(F/F'F+F'FJF'F'F'F &F'F'*&F+F'FOF'F)!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 712 5 "Here " } {XPPEDIT 18 0 "phi1" "6#%%phi1G" }{TEXT -1 2 " " }{TEXT 713 19 "is pr oportional to " }{XPPEDIT 18 0 "phi/(phi-1);" "6#*&%$phiG\"\"\",&F$F%F %!\"\"F'" }{TEXT -1 0 "" }{TEXT 714 33 ". The logarithmic derivative \+ of " }{TEXT -1 0 "" }{TEXT 715 0 "" }{XPPEDIT 18 0 "phi1;" "6#%%phi1G " }{TEXT -1 0 "" }{TEXT 716 78 " has similar properties. In particular , its numerator should be proportional " }{XPPEDIT 18 0 "Fi/gcd(Fi,di ff(Fi,x));" "6#*&%#FiG\"\"\"-%$gcdG6$F$-%%diffG6$F$%\"xG!\"\"" }{TEXT -1 0 "" }{TEXT 717 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "ldphi1: = factor( diff(phi1,x)/phi1 ):\nF0a:= collect( numer( ldphi1 ), x);\nG G:= collect( expand(F0a-4*(x^4+a*x^3+b*x^2+c*x+d)^2), x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$F0aG,4*&\"\"%\"\"\")%\"xG\"\")F(F(*&,&\"\"'! \"\"*&\"#?F(%\"pGF(F(F()F*\"\"(F(F(*&,.\"\"$F(*&F+F(%\"bGF(F(*&\"#MF(F 2F(F/*(\"#AF(F2F(%\"aGF(F/*&\"#!)F()F2\"\"#F(F(F>F(F()F*F.F(F(*&,0*(\" #[F(F2F(F>F(F(*&\"$c\"F(FAF(F/*(\"#UF(FAF(F>F(F(*&\"#;F(F9F(F/*&FMF(F2 F(F(*(\"#SF(F2F(F9F(F/*&\"#GF(%\"cGF(F(F()F*\"\"&F(F(*&,4*&\"#gF(%\"dG F(F(*(F;F(F2F(FSF(F/*(\"#\")F(FAF(F>F(F/*&\"#xF(FAF(F(*(FMF(FAF(F9F(F( *(\"#CF(F2F(F>F(F/*&\"#dF(FSF(F/*&\"\"*F(F9F(F(*(\"##)F(F2F(F9F(F(F()F *F'F(F(*&,2*(FPF(F2F(F9F(F/*(FPF(FAF(F>F(F(*&\"#IF(FSF(F(*(\"#oF(F2F(F SF(F(*(FBF(FAF(FSF(F(*(F'F(F2F(FZF(F/*(FioF(FAF(F9F(F/*&\"$A\"F(FZF(F/ F()F*F7F(F(*&,,*(\"#:F(FAF(F9F(F(*(F7F(FAF(FSF(F/*&\"#jF(FZF(F(*(\"#KF (F2F(FSF(F/*(F.F(F2F(FZF(F(F()F*FBF(F(**FBF(FAF(FSF(F*F(F(*&FAF(FZF(F( " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#GGG,4*&,(\"\"'!\"\"*&\"#?\"\"\" %\"pGF,F,*&\"\")F,%\"aGF,F)F,)%\"xG\"\"(F,F,*&,.\"\"$F,*&\"#!)F,)F-\" \"#F,F,*&\"#MF,F-F,F)*&\"\"%F,)F0F:F,F)F0F,*(\"#AF,F-F,F0F,F)F,)F2F(F, F,*&,2*(\"#SF,F-F,%\"bGF,F)*(\"#[F,F-F,F0F,F,*(F/F,F0F,FGF,F)*&\"$c\"F ,F9F,F)*&\"#;F,F-F,F,*&F+F,%\"cGF,F,*&FNF,FGF,F)*(\"#UF,F9F,F0F,F,F,)F 2\"\"&F,F,*&,8*(\"#\")F,F9F,F0F,F)*(\"#CF,F-F,F0F,F)*(F/F,F0F,FPF,F)*( FF,)FGF:F,F)*(FNF,F9F,FGF,F,F,)F2F>F,F,*&,6* (FFF,F9F,F0F,F,*&\"#IF,FPF,F,*(F:F,F9F,FPF,F,*(F[pF,F9F,FGF,F)*(\"#oF, F-F,FPF,F,*(FFF,F-F,FGF,F)*(F>F,F-F,F`oF,F)*(F/F,F0F,F`oF,F)*(F/F,FGF, FPF,F)*&\"$A\"F,F`oF,F)F,)F2F6F,F,*&,0*&\"#jF,F`oF,F,*&F>F,)FPF:F,F)*( F/F,FGF,F`oF,F)*(\"#KF,F-F,FPF,F)*(F(F,F-F,F`oF,F,*(\"#:F,F9F,FGF,F,*( F6F,F9F,FPF,F)F,)F2F:F,F,*&,&*(F/F,FPF,F`oF,F)*(F:F,F9F,FPF,F,F,F2F,F, *&F9F,F`oF,F,*&F>F,)F`oF:F,F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 718 15 "The expression " }{XPPEDIT 18 0 "GG;" "6#%#GGG" } {TEXT -1 1 " " }{TEXT 719 10 "should be " }{TEXT -1 1 "0" }{TEXT 721 25 ", so the coeeficients of " }{XPPEDIT 18 0 "GG;" "6#%#GGG" }{TEXT -1 1 " " }{TEXT 720 23 "give equations between " }{XPPEDIT 18 0 "a,b,c ,d,p;" "6'%\"aG%\"bG%\"cG%\"dG%\"pG" }{TEXT -1 0 "" }{TEXT 722 1 "." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "L:= [ coeffs(GG,x) ];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"LG7*,&*&)%\"pG\"\"#\"\"\"%\"dGF+F+*&\"\"%F+ )F,F*F+!\"\",&*(\"\")F+%\"cGF+F,F+F0*(F*F+F(F+F4F+F+,8*(\"#\")F+F(F+% \"aGF+F0*(\"#CF+F)F+F9F+F0*(F3F+F9F+F4F+F0*(\"#MF+F)F+F4F+F0*&\"#dF+F4 F+F0*(\"##)F+F)F+%\"bGF+F+*&\"\"*F+FCF+F+*&\"#_F+F,F+F+*&\"#xF+F(F+F+* &F.F+)FCF*F+F0*(\"#;F+F(F+FCF+F+,6*(\"#SF+F(F+F9F+F+*&\"#IF+F4F+F+*(F* F+F(F+F4F+F+*(FRF+F(F+FCF+F0*(\"#oF+F)F+F4F+F+*(FPF+F)F+FCF+F0*(F.F+F) F+F,F+F0*(F3F+F9F+F,F+F0*(F3F+FCF+F4F+F0*&\"$A\"F+F,F+F0,0*&\"#jF+F,F+ F+*&F.F+)F4F*F+F0*(F3F+FCF+F,F+F0*(\"#KF+F)F+F4F+F0*(\"\"'F+F)F+F,F+F+ *(\"#:F+F(F+FCF+F+*(\"\"$F+F(F+F4F+F0,.FdoF+*&\"#!)F+F(F+F+*&F>F+F)F+F 0*&F.F+)F9F*F+F0F9F+*(\"#AF+F)F+F9F+F0,2*(FPF+F)F+FCF+F0*(\"#[F+F)F+F9 F+F+*(F3F+F9F+FCF+F0*&\"$c\"F+F(F+F0*&FMF+F)F+F+*&\"#?F+F4F+F+*&FMF+FC F+F0*(\"#UF+F(F+F9F+F+,(F`oF0*&FfpF+F)F+F+*&F3F+F9F+F0" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 57 "Certainly within a minute we get the fo llowing solutions:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "GS:= Groebner [gsolve](L, [a,b,c,d,p] );\nSo:= map( Q->solve(convert(op(1,Q),set)), \+ GS );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#GSG<&7%7',&*&\"\"%\"\"\"% \"bGF+F+\"#:!\"\",&*&F*F+%\"aGF+F+\"#8F+,&%\"pGF+F+F+,&*&F*F+%\"dGF+F+ F+F.,&*&F*F+%\"cGF+F+\"\"(F+-%%plexG6'F:F7F4F1F,<\"7%7',&*&\"\"#F+F,F+ F+\"\"$F.,&F1F+FDF+,&*&FDF+F4F+F+F+F+,&*&\"#;F+F7F+F+F+F.,&*&FDF+F:F+F +F+F+F%#SoG<& <'/%\"bG#!\"&\"\"#/%\"cG#\"\"(F+/%\"aG!\"#/%\"pG#!\"\"F+/%\"dG#\"\"\" \"#;<'F0/F(#\"\"$F+/F-F5F3F7<'/F(\"\"!/F4FC/F-FC/F8FC/F1#!\"$\"\"%<'/F 4F6/F1#!#8FJ/F(#\"#:FJ/F-#!\"(FJ/F8#F:FJ" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 723 51 "Of all those solutions, only one is non-d egenerate:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "So:= convert(So, list ):\nfactor( map( subs, So, phi ) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7&,$*.\"#;\"\"\"%&alphaGF'%\"xGF',&F)F'F'!\"\"F',&*&\"\"#F'F)F'F'F'F+ \"\"),,*&F&F')F)\"\"%F'F'*&\"#KF')F)\"\"$F'F+*&\"#SF')F)F.F'F+*&\"#cF' F)F'F'F'F'!\"$F',$*,F&F'F(F'F)F'F*F'F,!\"%F',$**\"#kF'F(F'F*F',&*&F3F' F)F'F'F7F+F=F',$**FCF'F(F'F)F',&*&F3F'F)F'F'F'F+F=F'" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 43 "We distinguish the non-degenerate solution:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "nondegenerate:= proc( sbs, F, d \+ )\nlocal G; G:= factor(subs(sbs,F));\ndegree(numer(G))>=12 or degree(d enom(G))>=12\nend:\nK:= select( nondegenerate, So, phi, 12 );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG7#<'/%\"bG#!\"&\"\"#/%\"cG#\"\"( F+/%\"aG!\"#/%\"pG#!\"\"F+/%\"dG#\"\"\"\"#;" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 33 "To find the normalizing constant " }{XPPEDIT 724 0 "al pha;" "6#%&alphaG" }{TEXT -1 42 ", we have to find linear relation bet ween " }{XPPEDIT 725 0 "F0,F1,Fi;" "6%%#F0G%#F1G%#FiG" }{TEXT -1 22 ". \nSince the degree of " }{XPPEDIT 726 0 "F0;" "6#%#F0G" }{TEXT -1 60 " is lower, is enough to look at the leading coefficients of " } {XPPEDIT 727 0 "F1,Fi;" "6$%#F1G%#FiG" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(K[1], [F0,F1,Fi]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%*(%\"xG\"\"\",&F%F&F&!\"\"F&),&F%F&#F&\"\"#F(\"\")F&* $),0*&F,F&)F%\"\"'F&F(*&F3F&)F%\"\"&F&F&*&#\"#XF,F&*$)F%\"\"%F&F&F(*& \"#NF&)F%\"\"$F&F&*&#\"#**F-F&*$)F%F,F&F&F(*&#\"#LF-F&F%F&F(#F&\"#KF&F ,F&*$),,F:F&*&F,F&F?F&F(*&#F6F,F&FDF&F(*&#\"\"(F,F&F%F&F&#F&\"#;F&F@F& " }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 7 "We see " }{XPPEDIT 728 0 "al pha;" "6#%&alphaG" }{TEXT -1 41 " in an indirect way: we identify fi rst " }{XPPEDIT 729 0 "1-phi(x);" "6#,&\"\"\"F$-%$phiG6#%\"xG!\"\"" } {TEXT -1 4 " as " }{XPPEDIT 730 0 "F1/4/Fi;" "6#*(%#F1G\"\"\"\"\"%!\" \"%#FiGF'" }{TEXT -1 68 ", since this is obvious from comporisonof the leading coefficients. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "phi1i:= \+ factor( subs(K[1], F1/4/Fi) );\n'phi'=factor( 1-phi1i ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&phi1iG*(,(*&\"\"%\"\"\")%\"xG\"\"#F)F)*&F (F)F+F)!\"\"F)F.F,,,*&\"#;F))F+F(F)F)*&\"#KF))F+\"\"$F)F.*&\"$_\"F)F*F )F)*&\"$O\"F)F+F)F.F)F)F,,,*&F1F)F2F)F)*&F4F)F5F)F.*&\"#SF)F*F)F.*&\"# cF)F+F)F)F)F)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$phiG,$*,\"$K% \"\"\"%\"xGF(,&F)F(F(!\"\"F(,&*&\"\"#F(F)F(F(F(F+\"\"),,*&\"#;F()F)\" \"%F(F(*&\"#KF()F)\"\"$F(F+*&\"#SF()F)F.F(F+*&\"#cF(F)F(F(F(F(!\"$F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 731 329 "At the end we find th e same rational expression as in the argument of 'Hyp12a' ! This is ou r result of this computation. Since this solution is unique (up to fra ctional-linear transformations), and there is a composition of degree \+ 2,2,3 coverings with the target ramification data, the unique solution must be such a composition." }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "9 2 1 0" 6 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }