################################################  GAMMAs120 ############################################# 
#
Gammas120Version:= `GAMMAs120, Version 1.12 (1 April, 2004)`:
#
# by  R. Vidunas, Kyushu Univercity; Supported by the 21st Century COE Program
# This is a small Maple package for simplifying products and quotients of Gamma values.
# The package has to be read with this commando:
# > read "gammas120.mpl";
#
#
# Some combinations of Gamma values occur naturally as algebraic numbers or values of
# hypergeometric functions. They can be simplified using the classical recurrence equation,
# relfection and multiplication formulas. Maple does not use multiplication formulas.
# This package can simplify products and quotients of Gamma values GAMMA(x) at
# the rational points x such that the denominator of x divides 120.
# The expressions are products or quotients of the 16 "basic" gamma values at
# 1/3, 1/4, 1/5, 2/5, 1/8, 1/15, 1/20, 1/24, 1/40, 3/40, 7/40, 1/60, 7/60, 1/120, 7/120, 11/120
# and rational powers of Pi and several algebraic numbers.
# A few algebraic numbers are compactly denoted by psi[1], psi[-1], psi[2], psi[-2].
#
# Note that the Maple inline notation for the Gamma function is 'GAMMA', and
# that our considered Gamma values are simplified to the Gamma values in the interval (1/2,1).
# We use the inline notation 'Gamma', so that our 16 basic Gamma values won't be transformed 
# automatically. The Maple built-in facilities work only with the 'GAMMA' notation.
# Our basis routines are:
#       GammaSimplify(GT)
#       GAMMASimplify(GT).
# The result is a simplified Gamma term GT in our 'Gamma' or Maple's 'GAMMA' notation respectively.
# Also the trigonometric functions sin, cos, scs, sec with the argument k*Pi/120 are simplified.
#
# We provide the following lists of substitutions, to be used as arguments of 'subs':
#  Gammas:          for transforming all Gamma values in the interval (0,1) in terms of the 16 basic values;
#  Gamma2EllipticK: to express some basic Gamma values (in 'Gamma' notation) in terms of the EllipticK function; 
#  Gamma2GAMMA:     for changing our 'Gamma' notation to the Maple's 'GAMMA' notation;
#  GammaPsis:       for rewriting the constants psi[1], psi[-1], psi[2], psi[-2] as algebraic numbers.
#
# We also provide the following useful lists:
#  GammaValuesBasis:         the 16 basic Gamma values;
#  GammaNumbers:             the algebraic numbers used in our transformation formulas;
#  GammaAlgebraicIdentities: useful identites with those algebraic numbers for further by-hand simpification.
#
# The variables reserved by this packages are the mentioned ones and the following:
#       GammaHelp, psi[1], psi[-1], psi[2], psi[-2], Gammas120Version, GammaTrigoAdjust, GammaTempInterface 
#
# A related paper is: R. Vidunas, \'Expressions for values of the gamma function\',
# available at http://arxiv.org/math.CA/0403510
#
##########################################################################################################

GammaSimplify:= proc( T )
local Q;
  if type(T,`*`) then
    Q:= map( GammaTrigoAdjust, T )
  else  Q:= GammaTrigoAdjust(T)
  fi;
  subs(Gammas, Q)
end:

GAMMASimplify:= proc( T )
  subs( Gamma2GAMMA, GamaSimplify(T) )
end:

GammaTrigoAdjust:= proc( T )
local Q, k, QQ;
  Q:= op(1,T);
  if op(0,T)=`^` then
    GammaTrigoAdjust(Q)^op(2,T)
  elif op(0,T)=Gamma then   
    k:= floor(Q);
    QQ:= Q-k;
    pochhammer(QQ, k)*Gamma(QQ)
  elif op(0,T)=GAMMA then  Gamma(Q)
  else  Q:= Q/Pi;
    if type(Q,rational) and irem(120,denom(Q))=0 then
      if op(0,T)=csc then  Gamma(Q)*Gamma(1-Q)/Pi
      elif op(0,T)=sin then  Pi/Gamma(Q)/Gamma(1-Q)
      elif op(0,T)=sec then  Gamma(1/2+Q)*Gamma(1/2-Q)/Pi
      elif op(o,T)=cos then  Pi/Gamma(1/2+Q)/Gamma(1/2-Q)
      else T fi
    else T fi
  fi
end:

GammaPsis:= [ psi[1]=5+sqrt(5), psi[-1]=5-sqrt(5), psi[2]=sqrt(5+2*sqrt(5)), psi[-2]=sqrt(5-2*sqrt(5)) ]:

GammaValuesBasis:=  [ Gamma(1/3), Gamma(1/4), Gamma(1/5), Gamma(2/5), Gamma(1/8), Gamma(1/15), Gamma(1/20), Gamma(1/24),
  Gamma(1/40), Gamma(3/40), Gamma(7/40), Gamma(1/60), Gamma(7/60), Gamma(1/120), Gamma(7/120), Gamma(11/120) ]:

GammaNumbers:= [ Pi, 2, 3, 5, sqrt(2)+1, sqrt(3)+1, sqrt(3)+sqrt(2), sqrt(5)+sqrt(3), sqrt(6)+sqrt(5), sqrt(10)+3,
psi[1], sqrt(10)+sqrt(psi[1]), sqrt(10)+sqrt(psi[-1]), sqrt(15)+psi[2], sqrt(15)+psi[-2],
sqrt(psi[1])+sqrt(5), sqrt(psi[-1])+sqrt(5), sqrt(psi[1])+sqrt(3), sqrt(psi[-1])+sqrt(3) ]:

Gammas:= [Gamma(2/3)=2*Pi/sqrt(3)/Gamma(1/3),
Gamma(3/4)=Pi*sqrt(2)/Gamma(1/4),
Gamma(3/5)=Pi*2^(3/2)/sqrt(psi[1])/Gamma(2/5),
Gamma(4/5)=Pi*sqrt(2)*sqrt(psi[1])/sqrt(5)/Gamma(1/5),
Gamma(1/6)=sqrt(3)/2^(1/3)/sqrt(Pi)*Gamma(1/3)^2,
Gamma(5/6)=Pi^(3/2)*2^(4/3)/sqrt(3)/Gamma(1/3)^2,
Gamma(3/8)=sqrt(Pi)/sqrt(sqrt(2)+1)*Gamma(1/8)/Gamma(1/4),
Gamma(5/8)=sqrt(Pi)*2^(3/4)*Gamma(1/4)/Gamma(1/8),
Gamma(7/8)=Pi*2^(3/4)*sqrt(sqrt(2)+1)/Gamma(1/8),
Gamma(1/10)=sqrt(psi[1])/sqrt(Pi)/2^(7/10)*Gamma(1/5)*Gamma(2/5),
Gamma(3/10)=sqrt(Pi)*2^(7/5)*sqrt(5)/psi[1]*Gamma(1/5)/Gamma(2/5),
Gamma(7/10)=sqrt(Pi)*2^(3/5)*Gamma(2/5)/Gamma(1/5),
Gamma(9/10)=Pi^(3/2)*2^(7/10)*sqrt(psi[1])/sqrt(5)/Gamma(1/5)/Gamma(2/5),
Gamma(1/12)=3^(3/8)/sqrt(Pi)/2^(1/4)*sqrt(sqrt(3)+1)*Gamma(1/3)*Gamma(1/4),
Gamma(5/12)=sqrt(Pi)*2^(3/4)/3^(1/8)/sqrt(sqrt(3)+1)*Gamma(1/4)/Gamma(1/3),
Gamma(7/12)=sqrt(Pi)*2^(3/4)*3^(1/8)/sqrt(sqrt(3)+1)*Gamma(1/3)/Gamma(1/4),
Gamma(11/12)=Pi^(3/2)*2^(3/4)/3^(3/8)*sqrt(sqrt(3)+1)/Gamma(1/4)/Gamma(1/3),
Gamma(2/15)=sqrt(2)/3^(7/20)*5^(1/6)/sqrt(sqrt(15)+psi[-2])/Gamma(1/3)*Gamma(2/5)*Gamma(1/15),
Gamma(4/15)=sqrt(psi[1])*sqrt(2)/3^(3/10)/sqrt(sqrt(15)+psi[2])/sqrt(sqrt(15)+psi[-2])
/Gamma(1/5)*Gamma(2/5)*Gamma(1/15),
Gamma(7/15)=3^(9/20)*5^(1/3)/sqrt(psi[1])*sqrt(sqrt(15)+psi[-2])*Gamma(1/5)*Gamma(1/3)/Gamma(1/15),
Gamma(8/15)=Pi*4/3^(9/20)*5^(1/6)/sqrt(sqrt(15)+psi[2])/sqrt(psi[1])*Gamma(1/15)/Gamma(1/5)/Gamma(1/3),
Gamma(11/15)=2*Pi*3^(3/10)*Gamma(1/5)/Gamma(2/5)/Gamma(1/15),
Gamma(13/15)=Pi*sqrt(2)*3^(7/20)*sqrt(sqrt(15)+psi[2])/5^(1/6)*Gamma(1/3)/Gamma(2/5)/Gamma(1/15),
Gamma(14/15)=Pi/sqrt(2)/sqrt(5)*sqrt(psi[1])*sqrt(sqrt(15)+psi[2])*sqrt(sqrt(15)+psi[-2])/Gamma(1/15),
Gamma(3/20)=sqrt(Pi)*2^(19/20)*5^(1/8)/sqrt(psi[1])/sqrt(sqrt(10)+sqrt(psi[-1]))*Gamma(1/20)/Gamma(2/5),
Gamma(7/20)=sqrt(Pi)*2^(17/20)*5^(1/8)/sqrt(psi[1])/sqrt(sqrt(10)+sqrt(psi[1]))*Gamma(1/20)/Gamma(1/5),
Gamma(9/20)=Pi*2^(4/5)/sqrt(sqrt(10)+sqrt(psi[1]))/sqrt(sqrt(10)+sqrt(psi[-1]))
/Gamma(1/5)/Gamma(2/5)*Gamma(1/20),
Gamma(11/20)=2^(1/5)*sqrt(psi[1])*Gamma(1/5)*Gamma(2/5)/Gamma(1/20),
Gamma(13/20)=sqrt(Pi)*2^(23/20)*5^(3/8)/sqrt(psi[1])*sqrt(sqrt(10)+sqrt(psi[-1]))*Gamma(1/5)/Gamma(1/20),
Gamma(17/20)=sqrt(Pi)*2^(1/20)*sqrt(psi[1])*sqrt(sqrt(10)+sqrt(psi[1]))/5^(1/8)*Gamma(2/5)/Gamma(1/20),
Gamma(19/20)=Pi/sqrt(5)*sqrt(psi[1])*sqrt(sqrt(10)+sqrt(psi[1]))*sqrt(sqrt(10)+sqrt(psi[-1]))/Gamma(1/20),
Gamma(5/24)=sqrt(Pi)*2^(1/3)/sqrt(3)/sqrt(sqrt(2)+1)/sqrt(sqrt(3)+1)*Gamma(1/24)/Gamma(1/3),
Gamma(7/24)=sqrt(Pi)*2^(1/4)/3^(3/8)/sqrt(sqrt(3)+1)/sqrt(sqrt(3)+sqrt(2))/Gamma(1/4)*Gamma(1/24),
Gamma(11/24)=Pi*2^(1/12)/3^(3/8)/sqrt(sqrt(2)+1)/sqrt(sqrt(3)+sqrt(2))/Gamma(1/3)/Gamma(1/4)*Gamma(1/24),
Gamma(13/24)=2^(2/3)*3^(3/8)*sqrt(sqrt(3)+1)*Gamma(1/3)*Gamma(1/4)/Gamma(1/24),
Gamma(17/24)=2*sqrt(Pi)*3^(3/8)*sqrt(sqrt(2)+1)*Gamma(1/4)/Gamma(1/24),
Gamma(19/24)=sqrt(Pi)*2^(11/12)*sqrt(3)*sqrt(sqrt(3)+sqrt(2))*Gamma(1/3)/Gamma(1/24),
Gamma(23/24)=Pi*2^(3/4)*sqrt(sqrt(2)+1)*sqrt(sqrt(3)+1)*sqrt(sqrt(3)+sqrt(2))/Gamma(1/24),
Gamma(1/30)=3^(9/20)/sqrt(Pi)/2^(16/15)/5^(1/6)*sqrt(psi[1])*sqrt(sqrt(15)+psi[2])*Gamma(1/5)*Gamma(1/3),
Gamma(7/30)=3^(3/20)/sqrt(Pi)/2^(7/15)*5^(1/3)/sqrt(psi[1])*sqrt(sqrt(15)+psi[-2])*Gamma(1/3)*Gamma(2/5),
Gamma(11/30)=sqrt(Pi)*2^(23/30)/3^(1/20)*5^(1/6)/sqrt(sqrt(15)+psi[2])*Gamma(1/5)/Gamma(1/3),
Gamma(13/30)=sqrt(Pi)*2^(17/15)*3^(7/20)*5^(1/3)/sqrt(psi[1])/sqrt(sqrt(15)+psi[-2])*Gamma(1/3)/Gamma(2/5),
Gamma(17/30)=sqrt(Pi)*2^(41/30)/3^(7/20)*5^(1/6)/sqrt(sqrt(15)+psi[-2])/Gamma(1/3)*Gamma(2/5),
Gamma(19/30)=sqrt(Pi)*2^(11/15)*3^(1/20)/5^(1/6)*sqrt(psi[1])/sqrt(sqrt(15)+psi[2])*Gamma(1/3)/Gamma(1/5),
Gamma(23/30)=Pi^(3/2)*2^(59/30)/3^(3/20)*5^(1/6)/psi[1]*sqrt(sqrt(15)+psi[-2])/Gamma(2/5)/Gamma(1/3),
Gamma(29/30)=Pi^(3/2)/2^(13/30)/3^(9/20)/5^(5/6)*psi[1]*sqrt(sqrt(15)+psi[2])/Gamma(1/5)/Gamma(1/3),
Gamma(9/40)=Pi^(1/2)*2^(11/20)*5^(1/4)*sqrt(sqrt(10)+3)/sqrt(sqrt(2)+1)/sqrt(sqrt(10)+sqrt(psi[-1]))
*Gamma(1/8)^2*Gamma(1/20)/Gamma(1/40)*Gamma(7/40)/Gamma(2/5)/Gamma(3/40)/Gamma(1/4),
Gamma(11/40)=2^(1/5)*5^(1/8)*psi[1]^(1/4)/(sqrt(2)+1)^(3/4)*(sqrt(10)+sqrt(psi[1]))^(1/4)*(sqrt(10)+3)^(1/4)
*sqrt(sqrt(psi[1])+sqrt(5))/(sqrt(10)+sqrt(psi[-1]))^(1/4)*Gamma(1/8)^2*Gamma(1/5)*Gamma(7/40)/Gamma(1/4)/Gamma(3/40)/Gamma(1/40),
Gamma(13/40)=5^(1/4)/(sqrt(2)+1)^(1/4)/sqrt(sqrt(5)+sqrt(psi[-1]))*(sqrt(10)+sqrt(psi[-1]))^(1/4)
*(sqrt(10)+sqrt(psi[1]))^(1/4)*(sqrt(10)+3)^(1/4)*Gamma(1/5)*Gamma(7/40)/Gamma(1/20),
Gamma(17/40)=sqrt(psi[1])/2^(4/5)*(sqrt(10)+sqrt(psi[1]))^(1/4)*(sqrt(2)+1)^(1/4)/(sqrt(10)+3)^(1/4)
*(sqrt(10)+sqrt(psi[-1]))^(1/4)/sqrt(sqrt(5)+sqrt(psi[-1]))*Gamma(2/5)*Gamma(3/40)/Gamma(1/20),
Gamma(19/40)=sqrt(Pi)/2^(9/20)/5^(1/8)*psi[1]^(1/4)/(sqrt(2)+1)^(1/4)/(sqrt(10)+3)^(1/4)/sqrt(sqrt(psi[1])+sqrt(5))
*(sqrt(10)+sqrt(psi[1]))^(1/4)*(sqrt(10)+sqrt(psi[-1]))^(1/4)*Gamma(1/40)/Gamma(1/20),
Gamma(21/40)=sqrt(Pi)*2^(19/20)*Gamma(1/20)/Gamma(1/40),
Gamma(23/40)=Pi*2^(9/5)*5^(1/8)/sqrt(psi[1])/sqrt(sqrt(10)+sqrt(psi[-1]))*Gamma(1/20)/Gamma(2/5)/Gamma(3/40),
Gamma(27/40)=Pi*2^(3/2)*5^(1/8)/sqrt(psi[1])/sqrt(sqrt(10)+sqrt(psi[1]))*Gamma(1/20)/Gamma(1/5)/Gamma(7/40),
Gamma(29/40)=Pi*2^(4/5)/5^(1/4)*sqrt(sqrt(2)+1)/sqrt(sqrt(10)+sqrt(psi[1]))/sqrt(sqrt(10)+3)
*Gamma(1/4)*Gamma(1/40)*Gamma(3/40)/Gamma(1/8)^2/Gamma(1/5)/Gamma(7/40),
Gamma(31/40)=sqrt(Pi)*2^(9/20)/5^(1/8)*(1+sqrt(2))^(3/4)*psi[1]^(1/4)*(sqrt(10)+sqrt(psi[-1]))^(1/4)/(sqrt(10)+sqrt(psi[1]))^(1/4)
/(sqrt(10)+3)^(1/4)/sqrt(sqrt(psi[1])+sqrt(5))*Gamma(1/4)*Gamma(2/5)*Gamma(3/40)*Gamma(1/40)/Gamma(1/8)^2/Gamma(7/40)/Gamma(1/20),
Gamma(33/40)=Pi*2^(3/2)*5^(1/8)*(sqrt(2)+1)^(1/4)/sqrt(psi[1])/(sqrt(10)+sqrt(psi[1]))^(1/4)
*(sqrt(10)+sqrt(psi[-1]))^(1/4)/(sqrt(10)+3)^(1/4)*sqrt(sqrt(5)+sqrt(psi[-1]))/Gamma(7/40),
Gamma(37/40)=Pi*2/5^(1/8)/(sqrt(2)+1)^(1/4)*(sqrt(10)+sqrt(psi[1]))^(1/4)
/(sqrt(10)+sqrt(psi[-1]))^(1/4)*(sqrt(10)+3)^(1/4)*sqrt(sqrt(5)+sqrt(psi[-1]))/Gamma(3/40),
Gamma(39/40)=Pi*sqrt(2)/5^(3/8)*(sqrt(2)+1)^(1/4)*psi[1]^(1/4)*(sqrt(10)+sqrt(psi[1]))^(1/4)
*(sqrt(10)+sqrt(psi[-1]))^(1/4)*(sqrt(10)+3)^(1/4)*sqrt(sqrt(psi[1])+sqrt(5))/Gamma(1/40),
Gamma(11/60)=sqrt(Pi)*2^(5/4)/sqrt(3)*5^(7/24)/sqrt(psi[1])/sqrt(sqrt(15)+psi[2])/sqrt(sqrt(10)+sqrt(psi[1]))*Gamma(1/60)/Gamma(1/3),
Gamma(13/60)=sqrt(Pi)*2^(7/10)/3^(3/20)*5^(1/8)*sqrt(sqrt(3)+1)/sqrt(sqrt(5)+sqrt(3))/sqrt(sqrt(15)+psi[-2])*Gamma(7/60)/Gamma(2/5),
Gamma(17/60)=sqrt(Pi)*2^(3/4)/sqrt(3)*5^(1/24)*sqrt(psi[1])/sqrt(sqrt(15)+psi[-2])/sqrt(sqrt(10)+sqrt(psi[-1]))*Gamma(7/60)/Gamma(1/3),
Gamma(19/60)=sqrt(Pi)*2^(11/10)/3^(9/20)/5^(1/8)/sqrt(sqrt(3)+1)/sqrt(sqrt(5)+sqrt(3))/sqrt(sqrt(15)+psi[2])*Gamma(1/60)/Gamma(1/5),
Gamma(23/60)=Pi*2^(19/20)/3^(3/20)/5^(1/12)*sqrt(sqrt(3)+1)/sqrt(sqrt(5)+sqrt(3))/sqrt(sqrt(10)+sqrt(psi[-1]))*Gamma(7/60)/Gamma(2/5)/Gamma(1/3),
Gamma(29/60)=Pi*2^(17/20)/3^(9/20)/5^(1/12)/sqrt(sqrt(10)+sqrt(psi[1]))/sqrt(sqrt(3)+1)/sqrt(sqrt(5)+sqrt(3))*Gamma(1/60)/Gamma(1/5)/Gamma(1/3),
Gamma(31/60)=3^(9/20)/2^(1/10)/5^(1/6)*sqrt(psi[1])*sqrt(sqrt(15)+psi[2])*Gamma(1/5)*Gamma(1/3)/Gamma(1/60),
Gamma(37/60)=3^(3/20)*2^(3/10)*5^(1/3)/sqrt(psi[1])*sqrt(sqrt(15)+psi[-2])*Gamma(1/3)*Gamma(2/5)/Gamma(7/60),
Gamma(41/60)=sqrt(Pi)*2^(3/20)*3^(9/20)/5^(1/8)*sqrt(psi[1])*sqrt(sqrt(10)+sqrt(psi[1]))*Gamma(1/5)/Gamma(1/60),
Gamma(43/60)=sqrt(Pi)*2*sqrt(3)*5^(5/24)/sqrt(psi[1])/sqrt(sqrt(3)+1)*sqrt(sqrt(5)+sqrt(3))*Gamma(1/3)/Gamma(7/60),
Gamma(47/60)=sqrt(Pi)*2^(21/20)*3^(3/20)*5^(1/8)/sqrt(psi[1])*sqrt(sqrt(10)+sqrt(psi[-1]))*Gamma(2/5)/Gamma(7/60),
Gamma(49/60)=sqrt(Pi)*sqrt(3)/5^(1/24)*sqrt(psi[1])*sqrt(sqrt(3)+1)*sqrt(sqrt(5)+sqrt(3))*Gamma(1/3)/Gamma(1/60),
Gamma(53/60)=Pi*2^(5/4)*5^(1/4)/psi[1]/sqrt(sqrt(3)+1)*sqrt(sqrt(5)+sqrt(3))*sqrt(sqrt(15)+psi[-2])*sqrt(sqrt(10)+sqrt(psi[-1]))/Gamma(7/60),
Gamma(59/60)=Pi/2^(5/4)/5^(3/4)*psi[1]*sqrt(sqrt(3)+1)*sqrt(sqrt(5)+sqrt(3))*sqrt(sqrt(15)+psi[2])*sqrt(sqrt(10)+sqrt(psi[1]))/Gamma(1/60),
Gamma(13/120)=3^(7/40)/2^(53/120)/5^(1/8)*sqrt(10^(1/2)+psi[1]^(1/2))*sqrt(10^(1/2)+3)*psi[1]*(6^(1/2)+5^(1/2))^(1/4)
*(psi[1]^(1/2)+5^(1/2))^(1/4)*(1+3^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)/(1+2^(1/2))^(3/4)/(15^(1/2)+psi[-2])^(1/4)
/(10^(1/2)+psi[-1]^(1/2))^(1/4)/(5^(1/2)+psi[-1]^(1/2))^(1/4)/(5^(1/2)+3^(1/2))^(1/4)/(psi[1]^(1/2)+3^(1/2))^(1/4)
*Gamma(7/60)*Gamma(1/8)^2*Gamma(1/5)*Gamma(7/40)^2/Gamma(7/120)/Gamma(1/4)/Gamma(2/5)/Gamma(3/40)/Gamma(1/40),
Gamma(17/120)=(1+2^(1/2))/5^(1/8)/3^(1/4)/2^(1/30)*(10^(1/2)+psi[-1]^(1/2))^(1/4)*(15^(1/2)+psi[2])^(1/4)/psi[1]^(1/4)
/(10^(1/2)+psi[1]^(1/2))^(1/4)/sqrt(10^(1/2)+3)/sqrt(3^(1/2)+psi[-1]^(1/2))/sqrt(5^(1/2)+psi[-1]^(1/2))/(15^(1/2)+psi[-2])^(1/4)
*Gamma(2/5)*Gamma(3/40)^2*Gamma(11/120)*Gamma(7/120)*Gamma(1/120)*Gamma(1/4)/Gamma(1/8)^2/Gamma(1/60)/Gamma(1/5)/Gamma(7/40)^2,
Gamma(19/120)=3^(1/40)*2^(19/120)*psi[1]^(1/4)*(6^(1/2)+5^(1/2))^(1/4)*(10^(1/2)+psi[1]^(1/2))^(1/4)*(psi[1]^(1/2)+3^(1/2))^(1/4)
*(3^(1/2)+psi[-1]^(1/2))^(1/4)/(1+3^(1/2))^(1/4)/(1+2^(1/2))^(1/4)/(psi[1]^(1/2)+5^(1/2))^(1/4)/(15^(1/2)+psi[2])^(1/4)
/(5^(1/2)+psi[-1]^(1/2))^(1/4)/(5^(1/2)+3^(1/2))^(1/4)*Gamma(1/40)*Gamma(1/60)*Gamma(7/40)/Gamma(1/20)/Gamma(1/120),
Gamma(23/120)=Pi^(1/2)/2^(37/120)/3^(13/40)/5^(1/2)*(1+2^(1/2))^(3/4)*(15^(1/2)+psi[2])^(1/4)*(1+3^(1/2))^(1/4)/(10^(1/2)+3)^(1/2)
/(5^(1/2)+3^(1/2))^(1/4)/(6^(1/2)+5^(1/2))^(1/4)/(10^(1/2)+psi[1]^(1/2))^(1/4)/(psi[1]^(1/2)+3^(1/2))^(1/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4)
/(psi[1]^(1/2)+5^(1/2))^(1/4)/(5^(1/2)+psi[-1]^(1/2))^(1/4)*Gamma(7/120)*Gamma(1/4)*Gamma(1/40)*Gamma(3/40)*Gamma(11/120)*Gamma(1/120)
/Gamma(1/8)^2/Gamma(1/5)/Gamma(1/60)/Gamma(7/40)^2,
Gamma(29/120)=(1+2^(1/2))^(3/4)/2^(9/8)/3^(9/40)/5^(3/8)*psi[1]^(1/2)*(10^(1/2)+psi[-1]^(1/2))^(1/2)*(6^(1/2)+5^(1/2))^(1/4)
*(15^(1/2)+psi[2])^(1/4)/(10^(1/2)+psi[1]^(1/2))^(1/4)/(10^(1/2)+3)^(1/2)/(1+3^(1/2))^(1/4)/(5^(1/2)+3^(1/2))^(1/4)
/(psi[1]^(1/2)+3^(1/2))^(1/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4)/(psi[1]^(1/2)+5^(1/2))^(1/4)/(5^(1/2)+psi[-1]^(1/2))^(1/4)
*Gamma(1/4)*Gamma(1/40)*Gamma(3/40)^2*Gamma(2/5)*Gamma(11/120)/Gamma(1/8)^2/Gamma(1/5)/Gamma(7/40)/Gamma(1/20),
Gamma(31/120)=3^(9/40)/Pi^(1/2)/2^(97/60)/5^(7/24)*(1+2^(1/2))*psi[1]^(3/4)*sqrt(10^(1/2)+psi[-1]^(1/2))
*sqrt(15^(1/2)+psi[2])/sqrt(10^(1/2)+3)/sqrt(psi[1]^(1/2)+5^(1/2))/sqrt(5^(1/2)+psi[-1]^(1/2))*Gamma(1/3)
*Gamma(1/4)*Gamma(2/5)*Gamma(3/40)^2*Gamma(1/40)*Gamma(11/120)/Gamma(1/8)^2/Gamma(7/40)/Gamma(1/20)/Gamma(1/60),
Gamma(37/120)=5^(1/12)/3^(7/40)*(1+2^(1/2))^(1/2)/psi[1]*(10^(1/2)+psi[-1]^(1/2))^(1/4)*(15^(1/2)+psi[-2])^(1/4)*(15^(1/2)+psi[2])^(1/4)
/(10^(1/2)+psi[1]^(1/2))^(1/4)/sqrt(10^(1/2)+3)/sqrt(psi[1]^(1/2)+5^(1/2))/sqrt(3^(1/2)+psi[-1]^(1/2))*Gamma(1/3)*Gamma(1/4)
*Gamma(7/120)*Gamma(2/5)*Gamma(3/40)*Gamma(1/40)*Gamma(1/120)*Gamma(11/120)/Gamma(1/5)/Gamma(7/40)^2/Gamma(1/8)^2/Gamma(1/60)/Gamma(7/60),
Gamma(41/120)=3^(19/40)/2^(1/2)/5^(1/8)*psi[1]^(1/2)*sqrt(10^(1/2)+psi[1]^(1/2))*Gamma(1/5)*Gamma(1/40)*Gamma(7/40)/Gamma(1/120)/Gamma(1/20),
Gamma(43/120)=3^(1/4)/5^(5/12)/2^(7/8)*(15^(1/2)+psi[2])^(1/4)*(1+2^(1/2))^(3/4)*sqrt(10^(1/2)+psi[-1]^(1/2))*(5^(1/2)+3^(1/2))^(1/4)
/(1+3^(1/2))^(1/4)/(6^(1/2)+5^(1/2))^(1/4)/(10^(1/2)+psi[1]^(1/2))^(1/4)/(psi[1]^(1/2)+3^(1/2))^(1/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4)
/(psi[1]^(1/2)+5^(1/2))^(1/4)/(5^(1/2)+psi[-1]^(1/2))^(1/4)*Gamma(1/3)*Gamma(1/4)*Gamma(3/40)^2/Gamma(1/60)
*Gamma(1/120)*Gamma(7/120)*Gamma(11/120)*Gamma(2/5)/Gamma(7/40)^2/Gamma(7/60)/sqrt(10^(1/2)+3)/Gamma(1/8)^2/Gamma(1/5),
Gamma(47/120)=2^(1/5)*3^(13/40)*5^(1/4)*sqrt(10^(1/2)+psi[1]^(1/2))*sqrt(10^(1/2)+3)/sqrt(1+2^(1/2))
*Gamma(7/40)^2*Gamma(1/8)^2*Gamma(1/5)/Gamma(7/120)/Gamma(1/4)/Gamma(1/40)/Gamma(3/40),
Gamma(49/120)=3^(1/2)/2^(113/120)/5^(1/24)*psi[1]^(1/2)*(1+3^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)*(10^(1/2)+psi[1]^(1/2))^(1/4)
*(15^(1/2)+psi[2])^(1/4)*(5^(1/2)+3^(1/2))^(1/4)*(1+2^(1/2))^(1/4)/(psi[1]^(1/2)+3^(1/2))^(1/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4)
/(psi[1]^(1/2)+5^(1/2))^(1/4)/(5^(1/2)+psi[-1]^(1/2))^(1/4)*Gamma(1/3)*Gamma(11/120)/Gamma(1/60),
Gamma(53/120)=Pi^(1/2)*2^(119/120)*5^(1/2)*(15^(1/2)+psi[-2])^(1/4)*(10^(1/2)+psi[-1]^(1/2))^(1/4)*(psi[1]^(1/2)+5^(1/2))^(1/4)
*(5^(1/2)+3^(1/2))^(1/4)*(psi[1]^(1/2)+3^(1/2))^(1/4)/(1+3^(1/2))^(1/4)/psi[1]^(5/4)/(6^(1/2)+5^(1/2))^(1/4)
/(5^(1/2)+psi[-1]^(1/2))^(1/4)/(1+2^(1/2))^(1/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4)*Gamma(7/120)/Gamma(7/60),
Gamma(59/120)=Pi^(1/2)/2^(133/120)/5^(1/4)*psi[1]^(1/2)*(15^(1/2)+psi[2])^(1/4)*(10^(1/2)+psi[1]^(1/2))^(1/4)*(1+3^(1/2))^(1/4)
*(5^(1/2)+3^(1/2))^(1/4)/(6^(1/2)+5^(1/2))^(1/4)/(psi[1]^(1/2)+3^(1/2))^(1/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4)/(1+2^(1/2))^(1/4)
/(psi[1]^(1/2)+5^(1/2))^(1/4)/(5^(1/2)+psi[-1]^(1/2))^(1/4)*Gamma(1/120)/Gamma(1/60),
Gamma(61/120)=Pi^(1/2)*2^(59/60)*Gamma(1/60)/Gamma(1/120),
Gamma(67/120)=Pi^(1/2)*2^(53/60)*Gamma(7/60)/Gamma(7/120),
Gamma(71/120)=Pi*2^(31/15)/sqrt(3)*5^(7/24)/psi[1]^(1/2)/sqrt(15^(1/2)+psi[2])/sqrt(10^(1/2)+psi[1]^(1/2))*Gamma(1/60)/Gamma(1/3)/Gamma(11/120),
Gamma(73/120)=Pi*2^(77/40)/3^(13/40)*5^(1/4)/psi[1]*(1+2^(1/2))^(3/4)*(10^(1/2)+psi[-1]^(1/2))^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/4)
*(1+3^(1/2))^(1/4)*(psi[1]^(1/2)+3^(1/2))^(1/4)/sqrt(10^(1/2)+psi[1]^(1/2))/sqrt(10^(1/2)+3)/(psi[1]^(1/2)+5^(1/2))^(1/4)
/(5^(1/2)+3^(1/2))^(1/4)/(6^(1/2)+5^(1/2))^(1/4)/(15^(1/2)+psi[-2])^(1/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4)
*Gamma(1/4)*Gamma(3/40)*Gamma(1/40)*Gamma(7/120)/Gamma(7/40)^2/Gamma(1/8)^2/Gamma(1/5),
Gamma(77/120)=5^(1/6)*Pi*2^(3/2)/3^(1/4)*psi[1]^(3/4)*(10^(1/2)+psi[1]^(1/2))^(1/4)*sqrt(10^(1/2)+3)*sqrt(3^(1/2)+psi[-1]^(1/2))
*sqrt(5^(1/2)+psi[-1]^(1/2))/(1+2^(1/2))/(15^(1/2)+psi[2])^(1/4)/(15^(1/2)+psi[-2])^(1/4)/(10^(1/2)+psi[-1]^(1/2))^(3/4)*Gamma(1/8)^2
*Gamma(1/5)*Gamma(7/40)^2*Gamma(7/60)*Gamma(1/60)/Gamma(1/3)/Gamma(1/4)/Gamma(3/40)^2/Gamma(2/5)/Gamma(11/120)/Gamma(7/120)/Gamma(1/120),
Gamma(79/120)=Pi*2^(13/8)/3^(19/40)/5^(1/8)*(psi[1]^(1/2)+5^(1/2))^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/4)*(1+2^(1/2))^(1/4)/psi[1]^(1/4)
/(1+3^(1/2))^(1/4)/(5^(1/2)+3^(1/2))^(1/4)/(6^(1/2)+5^(1/2))^(1/4)/(10^(1/2)+psi[1]^(1/2))^(1/4)/(psi[1]^(1/2)+3^(1/2))^(1/4)
/(3^(1/2)+psi[-1]^(1/2))^(1/4)/(15^(1/2)+psi[2])^(1/4)*Gamma(1/20)*Gamma(1/120)/Gamma(1/40)/Gamma(1/5)/Gamma(7/40),
Gamma(83/120)=Pi*2^(15/8)*3^(7/40)*5^(5/12)*(1+3^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)*(10^(1/2)+psi[1]^(1/2))^(1/4)*
(psi[1]^(1/2)+3^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)*(psi[1]^(1/2)+5^(1/2))^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/4)*(10^(1/2)+3)^(1/2)
/(1+2^(1/2))^(3/4)/(15^(1/2)+psi[2])^(1/4)/(5^(1/2)+3^(1/2))^(1/4)/(10^(1/2)+psi[-1]^(1/2))^(1/2)*Gamma(1/60)*Gamma(1/5)*Gamma(7/40)^2
*Gamma(1/8)^2*Gamma(7/60)/Gamma(1/3)/Gamma(1/4)/Gamma(11/120)/Gamma(1/120)/Gamma(1/40)/Gamma(2/5)/Gamma(3/40)/Gamma(7/120),
Gamma(89/120)=Pi^(3/2)*2^(299/120)/3^(9/40)*5^(7/24)*sqrt(10^(1/2)+3)*(psi[1]^(1/2)+3^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)
*(psi[1]^(1/2)+5^(1/2))^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/4)/(1+3^(1/2))^(1/4)/psi[1]^(1/2)/(1+2^(1/2))^(3/4)/(6^(1/2)+5^(1/2))^(1/4)
/(10^(1/2)+psi[1]^(1/2))^(1/4)/(15^(1/2)+psi[2])^(1/4)/(5^(1/2)+3^(1/2))^(1/4)/sqrt(10^(1/2)+psi[-1]^(1/2))*Gamma(1/8)^2*Gamma(1/20)
*Gamma(7/40)*Gamma(1/60)/Gamma(1/40)/Gamma(2/5)/Gamma(3/40)^2/Gamma(1/4)/Gamma(1/3)/Gamma(11/120),
Gamma(91/120)=Pi*4*3^(9/40)*5^(1/8)*(10^(1/2)+3)^(1/2)*(psi[1]^(1/2)+5^(1/2))^(1/2)*(5^(1/2)+psi[-1]^(1/2))^(1/2)/psi[1]^(1/4)/(1+2^(1/2))
/sqrt(10^(1/2)+psi[-1]^(1/2))*Gamma(1/8)^2*Gamma(1/5)*Gamma(7/40)*Gamma(1/20)/Gamma(1/4)/Gamma(3/40)^2/Gamma(1/40)/Gamma(11/120)/Gamma(2/5),
Gamma(97/120)=Pi^(1/2)*2^(41/60)*3^(13/40)*5^(1/4)*psi[1]^(1/2)*(3^(1/2)+psi[-1]^(1/2))^(1/2)*(psi[1]^(1/2)+5^(1/2))^(1/2)*(10^(1/2)+3)^(1/2)
*(10^(1/2)+psi[1]^(1/2))^(1/4)*(15^(1/2)+psi[-2])^(1/4)/(1+2^(1/2))^(1/2)/(15^(1/2)+psi[2])^(1/4)/(10^(1/2)+psi[-1]^(1/2))^(1/4)
*Gamma(1/60)*Gamma(7/40)^2*Gamma(1/8)^2*Gamma(1/5)/Gamma(1/120)/Gamma(11/120)/Gamma(1/4)/Gamma(3/40)/Gamma(1/40)/Gamma(7/120),
Gamma(101/120)=Pi*2^(29/30)/3^(1/40)*Gamma(1/20)*Gamma(1/120)/Gamma(1/40)/Gamma(7/40)/Gamma(1/60),
Gamma(103/120)=4*Pi*5^(5/8)*2^(19/120)*3^(1/4)*(10^(1/2)+3)^(1/2)*(5^(1/2)+3^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)/psi[1]^(1/2)
*(10^(1/2)+psi[1]^(1/2))^(1/4)*(psi[1]^(1/2)+3^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)*(psi[1]^(1/2)+5^(1/2))^(1/4)
*(5^(1/2)+psi[-1]^(1/2))^(1/4)/(1+3^(1/2))^(1/4)/(1+2^(1/2))^(3/4)/(15^(1/2)+psi[2])^(1/4)/sqrt(10^(1/2)+psi[-1]^(1/2))
*Gamma(1/8)^2*Gamma(1/60)*Gamma(1/5)*Gamma(7/40)^2/Gamma(1/4)/Gamma(7/120)/Gamma(2/5)/Gamma(3/40)^2/Gamma(11/120)/Gamma(1/120),
Gamma(107/120)=2/15*3^(33/40)*Pi*2^(1/15)*5^(7/8)*(1+2^(1/2))^(1/2)*(10^(1/2)+psi[-1]^(1/2))^(1/2)/psi[1]^(1/2)/sqrt(10^(1/2)+psi[1]^(1/2))
*Gamma(1/40)*Gamma(2/5)*Gamma(3/40)*Gamma(1/4)*Gamma(7/120)/Gamma(1/5)/Gamma(7/40)^2/sqrt(10^(1/2)+3)/Gamma(1/8)^2/Gamma(7/60),
Gamma(109/120)=Pi*2^(9/8)*(1+3^(1/2))^(1/4)*(5^(1/2)+3^(1/2))^(1/4)*(psi[1]^(1/2)+3^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)/(6^(1/2)+5^(1/2))^(1/4)
*(psi[1]^(1/2)+5^(1/2))^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/4)/(10^(1/2)+psi[1]^(1/2))^(1/4)/(15^(1/2)+psi[2])^(1/4)/(1+2^(1/2))^(1/4)/Gamma(11/120),
Gamma(113/120)=Pi*2^(3/8)/5^(1/4)*psi[1]^(1/4)*(5^(1/2)+3^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)*(15^(1/2)+psi[-2])^(1/4)*(1+2^(1/2))^(1/4)
*(10^(1/2)+psi[-1]^(1/2))^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)/(1+3^(1/2))^(1/4)/(psi[1]^(1/2)+5^(1/2))^(1/4)
/(psi[1]^(1/2)+3^(1/2))^(1/4)/Gamma(7/120),
Gamma(119/120)=Pi/2^(1/8)/5^(1/2)*(1+3^(1/2))^(1/4)*psi[1]^(1/2)*(5^(1/2)+3^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)*(10^(1/2)+psi[1]^(1/2))^(1/4)
*(psi[1]^(1/2)+3^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)*(1+2^(1/2))^(1/4)*(psi[1]^(1/2)+5^(1/2))^(1/4)*(15^(1/2)+psi[2])^(1/4)
*(5^(1/2)+psi[-1]^(1/2))^(1/4)/Gamma(1/120),
Gamma(1/2)=sqrt(Pi) ]:

Gamma2GAMMA:= [ Gamma(1/3)=2/3*Pi*3^(1/2)/GAMMA(2/3), Gamma(1/4)=Pi*2^(1/2)/GAMMA(3/4),
Gamma(1/5)=Pi*2^(1/2)*psi[1]^(1/2)/5^(1/2)/GAMMA(4/5), Gamma(2/5)=Pi*2^(3/2)/psi[1]^(1/2)/GAMMA(3/5),
Gamma(1/8)=Pi*2^(3/4)*(1+2^(1/2))^(1/2)/GAMMA(7/8),
Gamma(1/15)=Pi/2^(1/2)/5^(1/2)*psi[1]^(1/2)*(15^(1/2)+psi[2])^(1/2)*(15^(1/2)+psi[-2])^(1/2)/GAMMA(14/15),
Gamma(1/20)=Pi/5^(1/2)*psi[1]^(1/2)*(10^(1/2)+psi[1]^(1/2))^(1/2)*(10^(1/2)+psi[-1]^(1/2))^(1/2)/GAMMA(19/20),
Gamma(1/24)=Pi*2^(3/4)*(1+2^(1/2))^(1/2)*(1+3^(1/2))^(1/2)*(3^(1/2)+2^(1/2))^(1/2)/GAMMA(23/24),
Gamma(1/40)=Pi*2^(1/2)/5^(3/8)*(1+2^(1/2))^(1/4)*psi[1]^(1/4)*(10^(1/2)+psi[1]^(1/2))^(1/4)*(10^(1/2)+psi[-1]^(1/2))^(1/4)*(10^(1/2)+3)^(1/4)
*(psi[1]^(1/2)+5^(1/2))^(1/2)/GAMMA(39/40),
Gamma(3/40)=2*Pi/5^(1/8)*(10^(1/2)+psi[1]^(1/2))^(1/4)*(10^(1/2)+3)^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/2)/(1+2^(1/2))^(1/4)
/(10^(1/2)+psi[-1]^(1/2))^(1/4)/GAMMA(37/40),
Gamma(7/40)=Pi*2^(3/2)*5^(1/8)*(1+2^(1/2))^(1/4)*(10^(1/2)+psi[-1]^(1/2))^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/2)/psi[1]^(1/2)
/(10^(1/2)+psi[1]^(1/2))^(1/4)/(10^(1/2)+3)^(1/4)/GAMMA(33/40),
Gamma(1/60)=Pi/2^(5/4)/5^(3/4)*psi[1]*(1+3^(1/2))^(1/2)*(5^(1/2)+3^(1/2))^(1/2)*(15^(1/2)+psi[2])^(1/2)*(10^(1/2)+psi[1]^(1/2))^(1/2)/GAMMA(59/60),
Gamma(7/60)=Pi*2^(5/4)*5^(1/4)*(5^(1/2)+3^(1/2))^(1/2)*(15^(1/2)+psi[-2])^(1/2)*(10^(1/2)+psi[-1]^(1/2))^(1/2)/psi[1]/(1+3^(1/2))^(1/2)/GAMMA(53/60),
Gamma(1/120)=Pi/2^(1/8)/5^(1/2)*(1+3^(1/2))^(1/4)*psi[1]^(1/2)*(5^(1/2)+3^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)*(10^(1/2)+psi[1]^(1/2))^(1/4)
*(psi[1]^(1/2)+3^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)*(1+2^(1/2))^(1/4)*(psi[1]^(1/2)+5^(1/2))^(1/4)*(15^(1/2)+psi[2])^(1/4)
*(5^(1/2)+psi[-1]^(1/2))^(1/4)/GAMMA(119/120),
Gamma(7/120)=Pi*2^(3/8)/5^(1/4)*psi[1]^(1/4)*(5^(1/2)+3^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)*(15^(1/2)+psi[-2])^(1/4)*(1+2^(1/2))^(1/4)
*(10^(1/2)+psi[-1]^(1/2))^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)/(1+3^(1/2))^(1/4)/(psi[1]^(1/2)+5^(1/2))^(1/4)
/(psi[1]^(1/2)+3^(1/2))^(1/4)/GAMMA(113/120),
Gamma(11/120)=Pi*2^(9/8)*(1+3^(1/2))^(1/4)*(5^(1/2)+3^(1/2))^(1/4)*(psi[1]^(1/2)+3^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)
*(psi[1]^(1/2)+5^(1/2))^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/4)/(6^(1/2)+5^(1/2))^(1/4)/(10^(1/2)+psi[1]^(1/2))^(1/4)/(15^(1/2)+psi[2])^(1/4)
/(1+2^(1/2))^(1/4)/GAMMA(109/120)
]:

GammaAlgebraicIdentities:= [(15^(1/2)+psi[2])*(15^(1/2)+psi[-2])=10/psi[1]*(psi[2]+sqrt(3))^2,
(15^(1/2)+psi[2])/(15^(1/2)+psi[-2])=(psi[-2]+sqrt(3))^2/4,
(10^(1/2)+psi[1]^(1/2))*(10^(1/2)+psi[-1]^(1/2))=(psi[2]+sqrt(5))^2,
(10^(1/2)+psi[1]^(1/2))/(10^(1/2)+psi[-1]^(1/2))=(psi[-2]+sqrt(5))^2/psi[1],
(psi[1]^(1/2)+5^(1/2))/(psi[-1]^(1/2)+5^(1/2))=2*sqrt(10)/psi[1]^(3/2)*(psi[-2]+sqrt(10)),
(psi[1]^(1/2)+5^(1/2))*(psi[-1]^(1/2)+5^(1/2))=psi[1]^(3/2)/2/sqrt(10)*(psi[2]+sqrt(10)),
(psi[1]^(1/2)+3^(1/2))*(psi[-1]^(1/2)+3^(1/2))=6^(1/2)*psi[2]+3+2*5^(1/2),
(psi[1]^(1/2)+3^(1/2))/(psi[-1]^(1/2)+3^(1/2))=6^(1/2)*psi[2]-4-5^(1/2),
(sqrt(15)+psi[2])*(sqrt(10)+sqrt(psi[1]))=2^(3/2)*5^(3/2)/psi[1]^2/(sqrt(3)+1)*(sqrt(5)+sqrt(3))*(psi[2]+sqrt(3)+2)^2,
(sqrt(10)+sqrt(psi[1]))/(sqrt(15)+psi[2])=5^(1/2)/2^(3/2)/psi[1]*(sqrt(3)+1)/(sqrt(5)+sqrt(3))*(psi[2]-sqrt(3)+2)^2,
(sqrt(15)+psi[-2])*(sqrt(10)+sqrt(psi[-1]))=psi[1]^2/2^(7/2)/5^(1/2)*(sqrt(3)+1)*(sqrt(5)+sqrt(3))*(psi[-2]-sqrt(3)+2)^2,
(sqrt(10)+sqrt(psi[-1]))/(sqrt(15)+psi[-2])=psi[1]/2^(5/2)/5^(1/2)/(sqrt(3)+1)/(sqrt(5)+sqrt(3))*(psi[-2]+sqrt(3)+2)^2,
2*sqrt(2)+sqrt(3)+1=2*(sqrt(2)+1)/(sqrt(3)+1)*(sqrt(3)+sqrt(2)),
2*sqrt(2)-sqrt(3)+1=(sqrt(2)+1)*(sqrt(3)+1)/(sqrt(3)+sqrt(2)),
2*sqrt(2)+sqrt(3)-1=(sqrt(3)+1)/(sqrt(2)+1)*(sqrt(3)+sqrt(2)),
2*sqrt(2)-sqrt(3)-1=2/(sqrt(2)+1)/(sqrt(3)+1)/(sqrt(3)+sqrt(2)),
2*sqrt(3)+sqrt(5)+1=2*sqrt(5)/psi[1]*(sqrt(3)+1)*(sqrt(5)+sqrt(3)),
2*sqrt(3)-sqrt(5)-1=8*sqrt(5)/psi[1]/(sqrt(3)+1)/(sqrt(5)+sqrt(3)),
2*sqrt(3)+sqrt(5)-1=psi[1]/sqrt(5)/(sqrt(3)+1)*(sqrt(5)+sqrt(3)),
2*sqrt(3)-sqrt(5)+1=psi[1]/sqrt(5)*(sqrt(3)+1)/(sqrt(5)+sqrt(3)),
1+2*sqrt(2)+sqrt(5)=2^(3/2)*5^(1/4)*sqrt(sqrt(2)+1)/sqrt(psi[1])*sqrt(sqrt(10)+3),
1+2*sqrt(2)-sqrt(5)=sqrt(2)/5^(1/4)*sqrt(sqrt(2)+1)*sqrt(psi[1])/sqrt(sqrt(10)+3),
1-2*sqrt(2)+sqrt(5)=2^(3/2)*5^(1/4)/sqrt(sqrt(2)+1)/sqrt(psi[1])/sqrt(sqrt(10)+3),
-1+2*sqrt(2)+sqrt(5)=sqrt(2)/5^(1/4)/sqrt(sqrt(2)+1)*sqrt(psi[1])*sqrt(sqrt(10)+3),
2+sqrt(2)+sqrt(3)+sqrt(5)=psi[1]^(3/2)/2/5^(3/4)*sqrt(sqrt(2)+1)*sqrt(sqrt(3)+sqrt(2))*sqrt(sqrt(5)+sqrt(3))/sqrt(sqrt(6)+sqrt(5)),
2-sqrt(2)+sqrt(3)+sqrt(5)=psi[1]^(3/2)/2/5^(3/4)/sqrt(sqrt(2)+1)/sqrt(sqrt(3)+sqrt(2))*sqrt(sqrt(5)+sqrt(3))*sqrt(sqrt(6)+sqrt(5)),
2+sqrt(2)-sqrt(3)+sqrt(5)=psi[1]^(3/2)/sqrt(2)/5^(3/4)*sqrt(sqrt(2)+1)/sqrt(sqrt(3)+sqrt(2))/sqrt(sqrt(5)+sqrt(3))*sqrt(sqrt(6)+sqrt(5)),
2-sqrt(2)-sqrt(3)+sqrt(5)=psi[1]^(3/2)/sqrt(2)/5^(3/4)/sqrt(sqrt(2)+1)*sqrt(sqrt(3)+sqrt(2))/sqrt(sqrt(5)+sqrt(3))/sqrt(sqrt(6)+sqrt(5)),
2+sqrt(2)+sqrt(3)-sqrt(5)=2^(5/2)*5^(3/4)/psi[1]^(3/2)*sqrt(sqrt(2)+1)*sqrt(sqrt(3)+sqrt(2))/sqrt(sqrt(5)+sqrt(3))*sqrt(sqrt(6)+sqrt(5)),
2-sqrt(2)+sqrt(3)-sqrt(5)=2^(5/2)*5^(3/4)/psi[1]^(3/2)/sqrt(sqrt(2)+1)/sqrt(sqrt(3)+sqrt(2))/sqrt(sqrt(5)+sqrt(3))/sqrt(sqrt(6)+sqrt(5)),
sqrt(3)+sqrt(5)-2-sqrt(2)=4*5^(3/4)/psi[1]^(3/2)*sqrt(sqrt(2)+1)/sqrt(sqrt(3)+sqrt(2))*sqrt(sqrt(5)+sqrt(3))/sqrt(sqrt(6)+sqrt(5)),
sqrt(2)+sqrt(3)+sqrt(5)-2=4*5^(3/4)/psi[1]^(3/2)/sqrt(sqrt(2)+1)*sqrt(sqrt(3)+sqrt(2))*sqrt(sqrt(5)+sqrt(3))*sqrt(sqrt(6)+sqrt(5)),
(3^(1/2)-2^(1/2))*psi[2]+2^(1/2)+1 = psi[1]^(7/4)/2^(11/8)/5*(15^(1/2)+psi[-2])^(1/4)*(10^(1/2)+psi[-1]^(1/2))^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/4)
*(1+3^(1/2))^(3/4)*(1+2^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)/(psi[1]^(1/2)+5^(1/2))^(1/4)/(5^(1/2)+3^(1/2))^(1/4)/(6^(1/2)+5^(1/2))^(1/4)
/(3^(1/2)+2^(1/2))^(1/2)/(psi[1]^(1/2)+3^(1/2))^(1/4),
(3^(1/2)-2^(1/2))*psi[2]+2^(1/2)-1=psi[1]^2/2^(5/8)/5*(5^(1/2)+3^(1/2))^(1/4)*(10^(1/2)+psi[-1]^(1/2))^(1/4)*(psi[1]^(1/2)+5^(1/2))^(1/4)
*(3^(1/2)+psi[-1]^(1/2))^(1/4)/(15^(1/2)+psi[-2])^(1/4)/(5^(1/2)+psi[-1]^(1/2))^(1/4)/(1+3^(1/2))^(3/4)/(1+2^(1/2))^(1/4)/(6^(1/2)+5^(1/2))^(1/4)
/(3^(1/2)+2^(1/2))^(1/2)/(psi[1]^(1/2)+3^(1/2))^(1/4),
(3^(1/2)+2^(1/2))*psi[2]-2^(1/2)-1 = 2^(7/8)/5^(1/4)*(1+2^(1/2))^(1/4)*psi[1]^(1/2)*(10^(1/2)+psi[-1]^(1/2))^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/4)
*(5^(1/2)+3^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)*(3^(1/2)+2^(1/2))^(1/2)*(psi[1]^(1/2)+3^(1/2))^(1/4)/(psi[1]^(1/2)+5^(1/2))^(1/4)
/(15^(1/2)+psi[-2])^(1/4)/(1+3^(1/2))^(3/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4),
(3^(1/2)+2^(1/2))*psi[2]-2^(1/2)+1 = 2^(1/8)/5^(1/4)*psi[1]^(1/4)*(1+3^(1/2))^(3/4)*(15^(1/2)+psi[-2])^(1/4)*(10^(1/2)+psi[-1]^(1/2))^(1/4)
*(psi[1]^(1/2)+5^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)*(3^(1/2)+2^(1/2))^(1/2)*(psi[1]^(1/2)+3^(1/2))^(1/4)/(5^(1/2)+3^(1/2))^(1/4)
/(5^(1/2)+psi[-1]^(1/2))^(1/4)/(1+2^(1/2))^(1/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4),
(3^(1/2)+2^(1/2))*psi[2]+2^(1/2)-1 = psi[1]^(9/4)/2^(9/8)/5*(5^(1/2)+psi[-1]^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)*(3^(1/2)+2^(1/2))^(1/2)
*(3^(1/2)+psi[-1]^(1/2))^(1/4)*(1+3^(1/2))^(3/4)/(15^(1/2)+psi[-2])^(1/4)/(10^(1/2)+psi[-1]^(1/2))^(1/4)
/(5^(1/2)+3^(1/2))^(1/4)/(psi[1]^(1/2)+5^(1/2))^(1/4)/(1+2^(1/2))^(1/4)/(psi[1]^(1/2)+3^(1/2))^(1/4),
(2^(1/2)-3^(1/2))*psi[2]+2^(1/2)+1 = 1/5*2^(3/8)*5^(3/4)*(1+3^(1/2))^(3/4)*psi[1]^(3/4)*(psi[1]^(1/2)+5^(1/2))^(1/4)*(1+2^(1/2))^(1/4)
*(psi[1]^(1/2)+3^(1/2))^(1/4)/(5^(1/2)+3^(1/2))^(1/4)/(5^(1/2)+psi[-1]^(1/2))^(1/4)/(6^(1/2)+5^(1/2))^(1/4)/(3^(1/2)+2^(1/2))^(1/2)
/(3^(1/2)+psi[-1]^(1/2))^(1/4)/(15^(1/2)+psi[-2])^(1/4)/(10^(1/2)+psi[-1]^(1/2))^(1/4),
(3^(1/2)+2^(1/2))*psi[2]+2^(1/2)+1 = 1/2^(7/8)/5*(1+2^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)*(3^(1/2)+2^(1/2))^(1/2)*(3^(1/2)+psi[-1]^(1/2))^(1/4)
*psi[1]^2*(15^(1/2)+psi[-2])^(1/4)*(5^(1/2)+3^(1/2))^(1/4)*(psi[1]^(1/2)+5^(1/2))^(1/4)/(5^(1/2)+psi[-1]^(1/2))^(1/4)
/(1+3^(1/2))^(3/4)/(10^(1/2)+psi[-1]^(1/2))^(1/4)/(psi[1]^(1/2)+3^(1/2))^(1/4),
(3^(1/2)-2^(1/2))*psi[2]-2^(1/2)+1 = 2^(5/8)/5^(1/4)*psi[1]^(1/2)*(5^(1/2)+psi[-1]^(1/2))^(1/4)*(15^(1/2)+psi[-2])^(1/4)
*(5^(1/2)+3^(1/2))^(1/4)*(psi[1]^(1/2)+3^(1/2))^(1/4)/((1+2^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)*(3^(1/2)+2^(1/2))^(1/2)
*(3^(1/2)+psi[-1]^(1/2))^(1/4)*(1+3^(1/2))^(3/4)*(10^(1/2)+psi[-1]^(1/2))^(1/4)*(psi[1]^(1/2)+5^(1/2))^(1/4)),
(3^(1/2)+2^(1/2))*psi[-2]+2^(1/2)+1 = 2^(17/8)*5^(1/4)*(3^(1/2)+2^(1/2))^(1/2)*(10^(1/2)+psi[1]^(1/2))^(1/4)*(psi[1]^(1/2)+3^(1/2))^(1/4)
*(3^(1/2)+psi[-1]^(1/2))^(1/4)*(1+2^(1/2))^(1/4)*(psi[1]^(1/2)+5^(1/2))^(1/4)*(15^(1/2)+psi[2])^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/4)
/(1+3^(1/2))^(3/4)/(5^(1/2)+3^(1/2))^(1/4)/psi[1]/(6^(1/2)+5^(1/2))^(1/4),
(3^(1/2)+2^(1/2))*psi[-2]+2^(1/2)-1 = 2^(15/8)*5^(3/4)*(5^(1/2)+3^(1/2))^(1/4)*(1+3^(1/2))^(3/4)*(3^(1/2)+2^(1/2))^(1/2)
*(10^(1/2)+psi[1]^(1/2))^(1/4)*(psi[1]^(1/2)+3^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)/(1+2^(1/2))^(1/4)/(psi[1]^(1/2)+5^(1/2))^(1/4)
/(15^(1/2)+psi[2])^(1/4)/(5^(1/2)+psi[-1]^(1/2))^(1/4)/psi[1]^(5/4)/(6^(1/2)+5^(1/2))^(1/4),
(2^(1/2)-3^(1/2))*psi[-2]+2^(1/2)+1 = 2^(15/8)*5^(1/2)*(1+2^(1/2))^(1/4)*(10^(1/2)+psi[1]^(1/2))^(1/4)*(psi[1]^(1/2)+5^(1/2))^(1/4)
*(5^(1/2)+psi[-1]^(1/2))^(1/4)*(1+3^(1/2))^(3/4)*(5^(1/2)+3^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)/psi[1]^(5/4)
/(3^(1/2)+2^(1/2))^(1/2)/(psi[1]^(1/2)+3^(1/2))^(1/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4)/(15^(1/2)+psi[2])^(1/4), 
(2^(1/2)-3^(1/2))*psi[-2]+2^(1/2)-1 = 2^(17/8)*5^(1/2)*(15^(1/2)+psi[2])^(1/4)*(10^(1/2)+psi[1]^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)
/(1+3^(1/2))^(3/4)/psi[1]/(5^(1/2)+3^(1/2))^(1/4)/(3^(1/2)+2^(1/2))^(1/2)/(psi[1]^(1/2)+3^(1/2))^(1/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4)
/(1+2^(1/2))^(1/4)/(psi[1]^(1/2)+5^(1/2))^(1/4)/(5^(1/2)+psi[-1]^(1/2))^(1/4),
(3^(1/2)-2^(1/2))*psi[-2]+2^(1/2)-1 = 2^(27/8)*5^(3/4)*(psi[1]^(1/2)+3^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)*(psi[1]^(1/2)+5^(1/2))^(1/4)
*(5^(1/2)+psi[-1]^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)/(10^(1/2)+psi[1]^(1/2))^(1/4)/(15^(1/2)+psi[2])^(1/4)/(1+3^(1/2))^(3/4)
/(5^(1/2)+3^(1/2))^(1/4)/psi[1]^(3/2)/(3^(1/2)+2^(1/2))^(1/2)/(1+2^(1/2))^(1/4),
(3^(1/2)+2^(1/2))*psi[-2]-2^(1/2)+1 = 2^(13/8)*5^(1/2)*(psi[1]^(1/2)+5^(1/2))^(1/4)*(5^(1/2)+psi[-1]^(1/2))^(1/4)*(15^(1/2)+psi[2])^(1/4)
*(1+3^(1/2))^(3/4)*(5^(1/2)+3^(1/2))^(1/4)*(3^(1/2)+2^(1/2))^(1/2)/(1+2^(1/2))^(1/4)/psi[1]^(5/4)/(10^(1/2)+psi[1]^(1/2))^(1/4)
/(psi[1]^(1/2)+3^(1/2))^(1/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4)/(6^(1/2)+5^(1/2))^(1/4),
(3^(1/2)-2^(1/2))*psi[-2]+2^(1/2)+1 = 2^(13/8)*(1+2^(1/2))^(1/4)*(5^(1/2)+3^(1/2))^(1/4)*(15^(1/2)+psi[2])^(1/4)*(1+3^(1/2))^(3/4)*5^(3/4)
*(psi[1]^(1/2)+3^(1/2))^(1/4)*(3^(1/2)+psi[-1]^(1/2))^(1/4)*(6^(1/2)+5^(1/2))^(1/4)/psi[1]^(5/4)/(psi[1]^(1/2)+5^(1/2))^(1/4)
/(5^(1/2)+psi[-1]^(1/2))^(1/4)/(10^(1/2)+psi[1]^(1/2))^(1/4)/(3^(1/2)+2^(1/2))^(1/2),
-(3^(1/2)+2^(1/2))*psi[-2]+2^(1/2)+1 = 5*2^(27/8)*(3^(1/2)+2^(1/2))^(1/2)*(1+2^(1/2))^(1/4)/(1+3^(1/2))^(3/4)/psi[1]^(3/2)
/(5^(1/2)+3^(1/2))^(1/4)/(psi[1]^(1/2)+3^(1/2))^(1/4)/(3^(1/2)+psi[-1]^(1/2))^(1/4)/(psi[1]^(1/2)+5^(1/2))^(1/4)
/(5^(1/2)+psi[-1]^(1/2))^(1/4)/(6^(1/2)+5^(1/2))^(1/4)/(10^(1/2)+psi[1]^(1/2))^(1/4)/(15^(1/2)+psi[2])^(1/4)
]:

Gamma2EllipticK:= [
 Gamma(1/3)=2^(7/9)*Pi^(1/3)/3^(1/12)*EllipticK((sqrt(3)-1)/2^(3/2))^(1/3),
 Gamma(1/4)=2*Pi^(1/4)*EllipticK(1/sqrt(2))^(1/2),
 Gamma(1/8)=Pi^(1/8)*2^(17/8)*EllipticK(1/sqrt(2))^(1/4)*EllipticK(sqrt(2)-1)^(1/2),
 Gamma(1/15)=Pi^(1/6)*2^(8/9)*3^(29/60)*5^(13/24)/sqrt(5+sqrt(5))*sqrt(sqrt(5+2*sqrt(5))+sqrt(3))*Gamma(1/5)^(1/2)/Gamma(2/5)^(1/2)
 *EllipticK((sqrt(3)-1)/2/sqrt(2))^(1/6)*EllipticK((3-sqrt(5))*(sqrt(5)-sqrt(3))*(2-sqrt(3))/2^(7/2))^(1/2),
 Gamma(1/20)=2^(9/40)*5^(1/8)/Pi^(1/4)*(5+sqrt(5))^(5/8)*sqrt(1+sqrt(5-2*sqrt(5)))
 *Gamma(1/5)^(1/2)*Gamma(2/5)^(1/2)*EllipticK(sqrt(1/2-sqrt(sqrt(5)-2)))^(1/2),
 Gamma(1/24)=2^(49/18)*3^(25/48)*Pi^(1/24)*sqrt(sqrt(2)+1)/(sqrt(3)+1)^(1/4)
 *EllipticK(1/sqrt(2))^(1/4)*EllipticK((sqrt(3)-1)/2^(3/2))^(1/6)*EllipticK((2-sqrt(3))*(sqrt(3)-sqrt(2)))^(1/2)
]:

Gammas120Version;
GammaTempInterface:= interface(prettyprint):
interface(prettyprint=1):
`by  R. Vidunas, Kyushu Univercity; Supported by the 21st Century COE Program`;
`This is a small Maple package for simplifying products and quotients of Gamma values.`;
`Please type \'GammaHelp()\' for more information, or see the starting comments in gammas120.mpl`;
interface(prettyprint=GammaTempInterface):

GammaHelp:= proc( )
interface(prettyprint=1);
print(`Some combinations of Gamma values occur naturally as algebraic numbers or values of`);
print(`hypergeometric functions. They can be simplified using the classical recurrence equation,`);
print(`relfection and multiplication formulas. Maple does not use multiplication formulas.`);
print(`This package can simplify products and quotients of Gamma values GAMMA(x) at`);
print(`the rational points x suc that the denominator of x divides 120.`);
print(`The expressions are products or quotients of the 16 basic gamma values`);
interface(prettyprint=GammaTempInterface);
print( op(GammaValuesBasis) ); interface(prettyprint=1);
print(`and rational powers of the following numbers`);
interface(prettyprint=GammaTempInterface);
print( op(GammaNumbers) );  interface(prettyprint=1);
print(` where`); interface(prettyprint=GammaTempInterface); print( op(GammaPsis) );
interface(prettyprint=1);
print(`Note that the Maple inline notation for the Gamma function is \'GAMMA\', and`);
print(`that relevant Gamma values are simplified to the Gamma values in the interval (1/2,1).`);
print(`We use the inline notation \'Gamma\', so that our 16 basic Gamma values won\'t be`); 
print(`changed automatically. The Maple built-in facilities work only with the \'GAMMA\' notation.`);
print(`Our basis routines are`);  interface(prettyprint=GammaTempInterface);
print( 'GammaSimplify(GT)', 'GAMMASimplify(GT)' ); interface(prettyprint=1);
print(`The result is a simplified Gamma term GT in our \'Gamma\' or Maples \'GAMMA\' notation respectively.`);
print(`Also the trigonometric functions sin, cos, scs, sec with argument k*Pi/120 are simplified.`);
print(`We also provide the following lists of substitutions, to be used as arguments of \'subs\':`);
interface(prettyprint=GammaTempInterface);
print( 'Gammas','Gamma2GAMMA','GammaPsis','Gamma2EllipticK' ); interface(prettyprint=1);
print(`They can be used respectively and (perhaps consequently) for the following pruposes:`);
print(`\'Gammas\': to transform all Gamma values in the interval (0,1) in terms of the 16 basic values;`);
print(`\'Gamma2EllipticK\': to express some basic Gamma values (in \'Gamma\' notation) in terms of the EllipticK function;`); 
print(`\'Gamma2GAMMA\': to change the \'Gamma\' notation to the Maple\'s \'GAMMA\' notation;`);
print(`\'GammaPsis\': to write down then algebraic numbers psi[1], psi[-1], psi[2], psi[-2] explicitly.`);
print(`There are also the following useful lists:`);  interface(prettyprint=1);
interface(prettyprint=GammaTempInterface);
print( 'GammaValuesBasis', 'GammaNumbers', 'GammaAlgebraicIdentities' );
interface(prettyprint=1); print(`which contain, respectively: the 16 basic Gamma values;`);
print(`the algebraic numbers used in our transformation formulas; and`);
print(`some useful formulas with those algebraic numbers for further by-hand simpification.`);
print(`The variables reserved by this packages are the mentioned ones and the following:`);
interface(prettyprint=GammaTempInterface);
print( 'GammaHelp()', 'psi[1]', 'psi[-1]', 'psi[2]', 'psi[-2]', 'Gammas120Version', 'GammaTrigoAdjust()', 'GammaTempInterface()');
interface(prettyprint=1);
print(`A related paper is: R. Vidunas, \'Expressions for values of the gamma function\',`);
print(`available at http://arxiv.org/math.CA/0403510`);
interface(prettyprint=GammaTempInterface);
end: