# This is a Maple package supplementing the papers 
# by Raimundas Vidunas and Galina Filipuk:
#
# [C] "A classification of coverings yielding 
#   Heun-to-hypergeometric reductions"
# available at http://arxiv.org/abs/1204.2730
#
# and
#
# [F] "Parametric transformations between the Heun 
#   and Gauss hypergeometric functions" 
# available at http://arxiv.org/abs/0910.3087v2
#
# The package contains:
# - the list HH of 48 Belyi functions in [C, Table 4]
# - the list PHR of 61 transformations P1-P61, encoding 
#   in particular the formulas of [H, Section 4.5]
#   (PHR = Parametric Heun Transformations)  
#
# To decode the formulas in PHR, use the routine
#   HeunReduction( PHR[k])
# with k a number from 1 to 61 
#
# To check the formulas, use the routine
#   # CheckSeries( Formula, … )
# where the other arguments are as in 
# the Maple routine   series( F, …)
#
# Package version: 2.0
# Date: February 4, 2013
# 

alias(omega=RootOf(Z^2+Z+1));

HeunReduction:= proc(Q)
   if Q[1]=[] then 
     error "No formula for this composite transformation is available yet"
   fi;
   HeunG(op(Q[1]),Q[2]) = Q[3]*Hypergeom(Q[4][1..2], [Q[4][3]], 
      factor(subs(y = HH[Q[5][1]], x = Q[5][3], Q[6..nops(Q)], Q[5][2])))
end:

CheckSeries:= proc(F) map(T->factor(evala(T)), series(op(1, F)-op(2, F), _rest)) end:

# Parametric Heun reductions

PHR:= [
[ [-1, 0, 2*a, 2*b, 2*c-1, a+b-c+1],
 x, 1, [a, b, c], [32, y, x] ],
[ [-1, 0, 4*a, 1+2*a-2*b, 2*a+2*b, 1+2*a-2*b],
 x, (1-x^2)^(-2*a), [a, b, a+b+1/2], [31, y, x] ],
[ [-1, 0, 4*a, 4*b, 2*a+2*b, a+b+1/2],
 x, 1, [a, b, a+b+1/2], [35, y, x] ],
[ [-1, 0, 6*a, 2/3-2*a, 4*a+2/3, 1/2],
 x, (1-2*x)^(-3*a)*(1+2*x)^(-3*a), [a, a+1/3, 2*a+5/6], [27, y, 2*x] ],
[ [-1, 4/3*a*(6*a-1), 4*a, 4*a+1/3, 2/3, 2*a+2/3],
 x, (1-2*x)^(-3*a), [a, a+1/3, 2/3], [46, y, (x-2)/(1-2*x)] ],
[ [-1, 0, 8*a, 1-4*a, 4*a+2/3, 2/3 ], x, (1-x)^(-a)*
 (1+x)^(-a)*(1-3*x)^(-3*a)*(1+3*x)^(-3*a), [a, a+1/3, 2*a+5/6], [19, y, 3*x] ],
[ [-1, 0, 8*a, 1-4*a, 1/3, 2*a+5/6], x, 
 (1-x)^(-a)*(1+x)^(-a), [a, 1/6-a, 2/3], [15, y, 3*x/2/sqrt(2)] ],
[ [-1, 0, 8*a, 8*a+2/3, 4*a+2/3, 6*a+1/2],
 x, (1+8*x^2)^(-3*a), [a, a+1/3, 2*a+5/6], [17, y, x] ],
[ [-1, 0, 4*a, 2*a+1/2, 2*a+1/2, 2*a+1/2],
 x, (1-I*x)^(-4*a), [a, a+1/4, 2*a+1/2], [48, 1-y, (x-I)/(x+I)] ],
[ [-1, 0, 8*a, 4*a+1/2, 4*a+1/2, 4*a+1/2],
 x, (1+x^2)^(-4*a), [a, a+1/4, 2*a+3/4], [41, 1/y, (x+I)/(x-I)] ],
[ [-1, 0, 6*a, 2*a+2/3, 4*a+1/3, 2*a+2/3], 
 x, (1-(omega+1)*x^2)^(-3*a), [a, a+1/3, 2*a+2/3], [28, y, (1+2*omega)*(x+1)/(x-1)] ],
[ [-1, 0, 12*a, 4*a+2/3, 8*a+1/3, 4*a+2/3],
 x, (1-x^2+x^4)^(-3*a), [a, a+1/3, 2*a+5/6], [5, y, x] ],
[ [-1, 0, 1, 8*a, 4*a+1/2, 2*a+3/4],
 x, (1-2*x^2)^(-4*a), [a, a+1/4, 2*a+3/4], [40, y, sqrt(2)*x] ],
[ [-1, 0, 12*a, 4/3-4*a, 4*a+2/3, 2*a+5/6],
 x, (1-16*x^2+16*x^4)^(-3*a), [a, a+1/3, 2*a+5/6], [2, y, 2*x] ],
[ [-3, 0, 3*a, 3*b, 2*a+2*b, 1/2],
 x, 1, [a, b, a+b+1/2], [34, y, (x+3)/4] ],
[ [], x, 1, [], [25, y, x] ],
[ [], x, 1, [], [12, y, x] ],
[ [], x, 1, [], [14, y, x] ],
[ [9, 18*a^2-9*a*b+6*a, 3*a, 2*a+b, a+b+1/3, 2*a-2*b+1],
 x, (1-x)^(-2*a), [a, b, a+b+1/3], [34, y/(y-1), x/(x+3)] ],
[ [9/8, 9*a^2+9*a*b-3/2*a, 4*a, 3*a+b, 3*a+3*b-1/2, a+b+1/2],
 x, (1-8/9*x)^(-3*a), [a, b, a+b+1/2], [47, y, 9*(x-1)/(8*x-9)] ],
[ [], x, 1, [], [25, y, x] ],
[ [], x, 1, [], [39, y, x] ],
[ [], x, 1, [], [20, y, x] ],
[ [], x, 1, [], [3, y, x] ],
[ [1/5, 2*a/3, 5*a, 1/2-a, 1/3, 1/2],
 x, (1-5*x)^(-2*a), [a,1/6-a,2/3], [30, y, (9*x-5)/4] ],
[ [25/9, 5*a/3, 6*a, 4*a+1/6, 1/3, 2/3],
 x, (1-9/25*x)^(-5*a), [a,1/6-a,2/3], [24, y, (3*x+125)/(9*x-25)] ],
[ [-80, -25*a*(8*a+1), 5*a, 5*a+1/4, 4*a+1/2, 1/2],
 x, (1-5*x/32)^(-4*a), [a,a+1/4,2*a+3/4], [29, y, 2*(x+80)/(32-5*x)] ],
[ [1/81, 50*a*(20*a+1)/81, 10*a, 5/6, 2/3, 2*a+5/6],
 x/9, (1-x/9)^(-a)*(1-9*x)^(-4*a), [a, 1/6-a, 2/3], [9, y, 2*(3*x+5)/(9*x-1)] ],
[ [2/27, 56*a/81, 8*a, 5/6-2*a, 1/3, 2*a+5/6],
 4/27*x, (1-4/27*x)^(-a)*(1-2*x)^(-2*a), [a, 1/6-a, 2/3], [16, y, x] ],
[ [27/28, a*(97-294*a)/24, 7*a, 2/3-a, 2/3, 1/2],
 27/16*x, (1-7/4*x)^(-3*a), [a, 1/6-a, 2/3], [23, y, x/(7*x-4)] ],
[ [32/5, 4*a/3, 5*a, 3*a+1/6, 1/3, 1/2],
 x, (1-5*x/32)^(-4*a), [a, 1/6-a, 2/3], [29, y, 2*(x+80)/(32-5*x)] ],
[ [27/32, 25*a*(11-30*a)/48, 10*a, 5/6, 2/3, 4*a+2/3],
 3*x/8, (1-3/8*x)^(-2*a)*(1-4/9*x)^(-3*a), [a, 1/6-a, 2/3], [10, y, x/(4*x-9)] ],
[ [3/128, 81*a*(51*a+1)/128, 9*a, 3*a+1/2, 2*a+5/6, 1/2],
 x/32, (1+27*x+6*x^2-x^3/2)^(-3*a), [a, a+1/3, 2*a+5/6], [13, y, x] ],
[ [-125/3, -30*a*(10*a+1), 6*a, 6*a+1/5, 4*a+2/5, 2*a+7/10],
 x, (1-9/25*x)^(-5*a), [a, a+1/5, 2*a+7/10], [24, y, (3*x+125)/(9*x-25)] ],
[ [32/81, 2*a*(179-686*a)/81, 10*a, 7/6-4*a, 2/3, 2*a+5/6],
 4/27*x, (1-4/27*x)^(-a)*(1-3/8*x)^(-2*a), [a, 1/6-a, 2/3], [8, y, x-4] ],
[ [125/189, 8*a*(38-147*a)/81, 7*a, 5/6-3*a, 2/3, 1/2],
 4/27*x, (1-28/125*x)^(-2*a), [a, 1/6-a, 2/3], [22, y, x] ],
[ [17+12*sqrt(2), 2*a*(8*a+1)/(3-2*sqrt(2)),4*a, 4*a+1/2, 1/2, 4*a+1/2],
 x/(3-2*sqrt(2)), (1+x)^(-4*a), [a, a+1/4, 1/2], [35, y, (x-1)/(x+1)] ],
[ [4*sqrt(3)-7, 3*a*(12*a+1)/(3+2*sqrt(3)), 6*a, 6*a+1/2, 1/2, 6*a+1/2],
 x/(3+2*sqrt(3)), (1-2*x-x^2/3)^(-3*a), [a, a+1/3, 1/2], [28, y, x] ],
[ [97+56*sqrt(3), 9*a*(24*a+1)/(14-8*sqrt(3)), 6*a, 6*a+1/4, 3/4, 6*a+1/4],
 x/(-7+4*sqrt(3)), (1+14*x+x^2)^(-3*a), [a, 1/4-a, 3/4], [43, y, 2*(x+1)/(x-1)] ],
[ [15*sqrt(3)-26, 8*a*(12*a+1)/(3*(5+3*sqrt(3))), 4*a, 4*a+1/3, 2/3, 4*a+1/3],
 x/(5+3*sqrt(3)), (1-2*x)^(-3*a), [a, a+1/3, 2/3], [47, y, 9/4*x/(2*x-1)] ],
[ [], x, 1, [], [20, y, x] ],
[ [9+4*sqrt(5), 5*a*(1+10*a)/(10-4*sqrt(5)), 5*a, 3*a+3/10, 1/2, 4*a+2/5],
 5*x/(5-2*sqrt(5)), (1+x)^(-5*a), [a, a+1/5, 1/2], [45, y, 4*x/(x+1)] ],
[ [(-123+55*sqrt(5))/2, 12*a*(1+60*a)/(11+5*sqrt(5)), 12*a, 2*a+5/6, 10*a+1/6, 2*a+5/6],
 2*x/(11+5*sqrt(5)), (1+12*x+14*x^2-12*x^3+x^4)^(-3*a), [a, a+1/3, 2*a+5/6], [4, y, x] ],
[ [(-7+24*I)/25, 5*a*(40*a-1)/(6-8*I), 6*a, 4*a+1/4, 10*a-1/4, 3/4], 5*x/(-3+4*I), 
 (1+x/5)^(-4*a)*(1+6/5*x+x^2)^(-a), [a, a+1/4, 2*a+3/4], [42, 1/y, (x+5)/2/x] ],
[ [(117+44*I)/125, a*(49-228*a)/(11-2*I), 6*a, 5/6-4*a, 2*a+5/6, 1/2],
 x/(-11+2*I), (1+2*x+x^2/5)^(-3*a), [a, a+1/3, 2*a+5/6], [26, 1/y, (x+5)/2] ],
[ [(117+44*I)/125, 10*a*(40*a-1)/(11-2*I), 6*a, 6*a+1/5, 8*a-1/5, 2*a+7/10],
 25*x/(11-2*I), (1-2*x)^(-5*a), [a, a+1/5, 2*a+7/10], [42, y, x/(2*x-1)] ],
[ [(23+10*sqrt(-2))/27, 32/3*a*(1-6*a)/(5-sqrt(-2)), 4*a, 2/3-4*a, 2/3, 1/2],
 x/(5-sqrt(-2)), 1, [a, 1/6-a, 2/3], [36, y, (x-4)/3] ],
[ [(17+56*sqrt(-2))/81, a*(17-40*a)/(7-4*sqrt(-2)), 4*a, 3/4-2*a, 2*a+3/4, 1/2],
 x/(-7+4*sqrt(-2)), (1+x/3)^(-4*a), [a, a+1/4, 2*a+3/4], [36, y, -4/(x+3)] ],
[ [(17+56*sqrt(-2))/81, 4*a*(13-64*a)/(7-4*sqrt(-2)), 10*a, 4/3-6*a, 2/3, 2*a+5/6],
 3*x/(7-4*sqrt(-2)), (1-14/27*x+x^2/9)^(-a), [a, 1/6-a, 2/3], [7, y, x-2] ],
[ [(241+22*sqrt(-2))/243, 8*a*(7-8*a)/(22-sqrt(-2)), 4*a, 5/6, 2/3, 2*a+7/12],
 18*x/(22-sqrt(-2)), (1-x)^(-4*a), [a, a+1/4, 2/3], [36, y, (x-4/3)/(1-x)] ],
[ [-omega,3*(1-omega)*a*b,3*a,3*b,a+b+1/3,a+b+1/3],
 x, 1, [a,b,a+b+1/3], [33,1-y,(1-omega)*x+omega] ],
[ [], x, 1, [], [38, y, x] ],
[ [], x, 1, [], [6, y, x] ],
[ [], x, 1, [], [38, y, x] ],
[ [], x, 1, [], [1, y, x] ],
[ [(55+39*omega)/49, 2*a*(71-348*a)/(3*(5-3*omega)), 8*a, 7/6-6*a, 2*a+5/6, 2/3],
 x/(5-3*omega), (1+10*x-9*x^2+2*x^3-x^4/7)^(-2*a), [a, a+1/2, 2*a+5/6], [18, 1/(1-y), -x] ],
[ [(-87+91*sqrt(-7))/256, 2*a*(31-147*a)/(13-7*sqrt(-7)), 9*a, 7/6-5*a, 1/2, 2*a+5/6],
 8*x/(13-7*sqrt(-7)), (1-13/32*x+x^2/8)^(-a), [a, 1/6-a, 1/2], [11, 1-y, -x] ],
[ [(781+171*sqrt(-15))/1024, 10*a*(23-45*a)/(95-9*sqrt(-15)), 5*a, 9/10-a, 1/2, 2*a+7/10],
 20*x/(95-9*sqrt(-15)), (1-x/4)^(-5*a), [a, a+1/5, 1/2], [37, 1-y, 3/(4-x)] ],
[ [(-7+33*sqrt(-15))/128, 25*a*(6*a-1)/(3*sqrt(-15)-11), 5*a, 5/6-5*a, 1/2, 2/3],
 6*x/(11-3*sqrt(-15)), 1, [a, 1/6-a, 1/2], [37, 1-y, 3/4*(1-x)] ],
[ [1+2*I, a*(5/4+(7+24*I)*a), 5*a, 3*a+1/4, 3/4, 8*a], 
 x, (1-x)^(-4*a), [a, 1/4-a, 3/4], [44, y, x/(1+2*I)] ],
[ [(3-12*omega)/7, 2*a*(7+2*a*(omega-18))/(3+omega), 7*a, 1-5*a, 2/3, 1/2],
 x/(1+2*omega), (1-(1-2*omega)/9*x)^(-a), [a, 1/6-a, 2/3], [21, y, x/(1+2*omega)+1],
 omega=-1-omega ]
]:

HH:=
[ 64*x^3*(x^3-1)^3/(8*x^3-9),
27/4*x^2*(x^2-4)/(x^4-4*x^2+1)^3,
4/27*x^3*(x^3-6*x+6)^3/(x-1)^3/(2*x-3)^2/(x+3),
1728*x^5*(x^2-11*x-1)/(x^4-12*x^3+14*x^2+12*x+1)^3,
27/4*x^4*(x^2-1)^2/(x^4-x^2+1)^3,
-64*x^3*(x^3-1)^3/(8*x^3+1)^3,
-4/27*(x+2)*(x^3+3*x+2)^3/(3*x^2-2*x+11),
4/27*(x+4)*(x^3-6*x-2)^3/(3*x+4)^2/(4*x-11),
-(x+10)/27*(4*x^3-15*x+10)^3/(5*x-4)/(3*x-2)^5,
# x/64*(9*x^3-90*x^2+105*x+40)^3/(x-9)/(9*x-1)^4,
4*x*(9*x^3-20*x^2+10*x+10)^3/(5*x-8)^2/(4*x-1)^5,
4/27*(x^3+4*x^2+10*x+6)^3/(4*x^2+13*x+32),
27/4*x^2*(x-3)/(x^3-3*x^2+1)^3,
27/4*x*(4*x-3)^5/(x^3-12*x^2-54*x-2)^3,
27/4*x^3*(3*x+4)^2/(x^3-3*x-4)^3,
64*x^2*(x^2-1)^3/(8*x^2-9),
4/27*x^2*(x^2-8*x+10)^3/(4*x-27)/(2*x-1)^2,
-64*x^2*(x^2-1)^3/(8*x^2+1)^3,
(x^2+13*x+49)*(x^2+5*x+1)^3/1728/x,
-64*x^2/(x^2-1)^3/(x^2-9),
16*x^3*(2*x+1)*(x-4)/(x^2-2*x-2)^4,
4*(x-1)*((1+2*omega)*x^2-3*x-omega)^3/(4-(1+3*omega)*x),
4/27*x*(4*x^2-35*x+70)^3/(28*x-125)^2,
x/4*(9*x^2-14*x-7)^3/(7*x-1)^4,
x^3/64*(x+5)^2*(x+8)/(3*x-1),
-4*x^3*(x-1)^2*(x+2)/(3*x-2)^2,
(x^2-5)^3/27/(2*x-5),
27/4*x^2/(x^2-1)^3,
36*x*(x^2+3)^2/(x^2+6*x-3)^3,
4/27*x^3*(x-5)^2/(5*x+2),
x^3*(4*x+5)^2/(5*x+4)^2,
-4*x^2/(x^2-1)^2,
x^2, x^3, x*(4*x-3)^2, 
4*x^2*(1-x^2), -x^3*(3*x+4),
4/27*x^3*(4*x^2+5*x+10),
4*x^3*(1-x^3), x^2*(4*x^2-3)^2,
4*x^2*(x^2-2)/(x^2-1)^4,
-4*x^4/(x^4-1)^2,
x^4/4*(x^2+2*x+5)/(2*x-1),
27/4*(x^2-4)/(x^2-3)^3,
x*(x-1-2*I)^4/((1+2*I)*x-1)^4,
x/4*(x^2-5*x+5)^2,
x^3*(x+2)/(2*x+1),
64*x*(x-1)^3/(8*x-9), x^4 ]: