Ly := y*(y-1)^2*(1+y)^2*(2*y*z-z^2-1)*(y^3*z-3*y*z+z^2+1)*(8*y^3*z-6*y*z-z^2-1)*(8*y^3*z+3*y^2*z^2+3*y^2-7*z^2-7)*Dy^4+(y-1)*(1+y)*(1664*y^12*z^4-276*y^10*z^6-7464*y^10*z^4+44*y^9*z^5+1818* y^8*z^6-336*y^7*z^7-276*y^10*z^2+44*y^9*z^3+8244*y^8*z^4+5772*y^7*z^5-6018*y^6*z^6+1116*y^5*z^7+21*y^4*z^8+1818*y^8*z^2+5772*y^7*z^3-12036*y^6*z^4-2952*y^5*z^5+3978*y^4*z^6-544*y^3*z^ 7-54*y^2*z^8-336*y^7*z-6018*y^6*z^2-2952*y^5*z^3+7914*y^4*z^4-876*y^3*z^5-594*y^2*z^6+84*y*z^7-7*z^8+1116*y^5*z+3978*y^4*z^2-876*y^3*z^3-1080*y^2*z^4+252*y*z^5-28*z^6+21*y^4-544*y^3*z -594*y^2*z^2+252*y*z^3-42*z^4-54*y^2+84*y*z-28*z^2-7)*Dy^3+(5888*y^13*z^4+840*y^12*z^5-834*y^11*z^6+840*y^12*z^3-30396*y^11*z^4-2923*y^10*z^5+7332*y^9*z^6-1371*y^8*z^7-834*y^11*z^2-\ 2923*y^10*z^3+40224*y^9*z^4+29865*y^8*z^5-28527*y^7*z^6+5331*y^6*z^7+30*y^5*z^8+7332*y^9*z^2+29865*y^8*z^3-60582*y^7*z^4-23043*y^6*z^5+26469*y^5*z^6-4339*y^4*z^7-108*y^3*z^8-1371*y^8* z-28527*y^7*z^2-23043*y^6*z^3+52662*y^5*z^4-4395*y^4*z^5-6099*y^3*z^6+849*y^2*z^7+14*y*z^8+5331*y^6*z+26469*y^5*z^2-4395*y^4*z^3-11982*y^3*z^4+3114*y^2*z^5-133*y*z^6+42*z^7+30*y^5-\ 4339*y^4*z-6099*y^3*z^2+3114*y^2*z^3-294*y*z^4+126*z^5-108*y^3+849*y^2*z-133*y*z^2+126*z^3+14*y+42*z)*Dy^2+(5920*y^12*z^4+1872*y^11*z^5-669*y^10*z^6+1872*y^11*z^3-30666*y^10*z^4-5726* y^9*z^5+8031*y^8*z^6-1554*y^7*z^7-669*y^10*z^2-5726*y^9*z^3+30702*y^8*z^4+39138*y^7*z^5-31836*y^6*z^6+5730*y^5*z^7+6*y^4*z^8+8031*y^8*z^2+39138*y^7*z^3-63528*y^6*z^4-10806*y^5*z^5+ 18792*y^4*z^6-3086*y^3*z^7-24*y^2*z^8-1554*y^7*z-31836*y^6*z^2-10806*y^5*z^3+38004*y^4*z^4-9042*y^3*z^5-855*y^2*z^6+126*y*z^7-14*z^8+5730*y^5*z+18792*y^4*z^2-9042*y^3*z^3-1662*y^2*z^4 +756*y*z^5-119*z^6+6*y^4-3086*y^3*z-855*y^2*z^2+756*y*z^3-210*z^4-24*y^2+126*y*z-119*z^2-14)*Dy+3*(360*y^11*z^3+201*y^10*z^4-27*y^9*z^5+201*y^10*z^2-2030*y^9*z^3-540*y^8*z^4+591*y^7*z ^5-108*y^6*z^6-27*y^9*z-540*y^8*z^2+1590*y^7*z^3+3054*y^6*z^4-2385*y^5*z^5+390*y^4*z^6+591*y^7*z+3054*y^6*z^2-4842*y^5*z^3+366*y^4*z^4+777*y^3*z^5-108*y^2*z^6-108*y^6-2385*y^5*z+366*y ^4*z^2+1554*y^3*z^3-639*y^2*z^4+84*y*z^5-14*z^6+390*y^4+777*y^3*z-639*y^2*z^2+168*y*z^3-42*z^4-108*y^2+84*y*z-42*z^2-14)*z ; # Decomposes as a symmetric product over the following extension: ( 3*(2*y*z-z^2-1)*(8*y^3*z-6*y*z-z^2-1) )^(1/2); _Envdiffopdomain := [Dy,y]; # Ly is a pullback of a smaller operator under this pullback: x = (y^2-1)/(2*y^3*z-z^2-1) namely: Ly_reduced_to_x := Dx^4+1/2*(2268*z^8*x^5+6804*z^6*x^5-486*z^6*x^4+6804*z^4*x^5-972*z^4*x^4+2268*z^2*x^5-3942*z^4*x^3-486*z^2*x^4+39*z^4*x^2-3942*z^2*x^3+2238*z^2*x^2+36*x*z^2+39*x^2+36*x-40) /(27*z^4*x^3+27*z^2*x^3-27*z^2*x^2+1)/(3*x*z^2+3*x+4)/(x*z^2+x-1)/x*Dx^3+1/4*(17010*z^8*x^5+51030*z^6*x^5+4050*z^6*x^4+51030*z^4*x^5+8100*z^4*x^4+17010*z^2*x^5-28701*z^4*x^ 3+4050*z^2*x^4+102*z^4*x^2-28701*z^2*x^3+10848*z^2*x^2+168*x*z^2+102*x^2+168*x-64)/(27*z^4*x^3+27*z^2*x^3-27*z^2*x^2+1)/(3*x*z^2+3*x+4)/(x*z^2+x-1)/x^2*Dx^2+3/4*(5670*x^4*z ^8+17010*z^6*x^4+3915*z^6*x^3+17010*z^4*x^4+7830*z^4*x^3+5670*z^2*x^4-8586*z^4*x^2+3915*z^2*x^3+6*x*z^4-8586*z^2*x^2+2016*x*z^2+16*z^2+6*x+16)/x^2/(3*x*z^2+3*x+4)/(27*z^4*x ^3+27*z^2*x^3-27*z^2*x^2+1)/(x*z^2+x-1)*Dx+9/16*(945*z^6*x^3+2835*z^4*x^3+1080*z^4*x^2+2835*z^2*x^3+2160*z^2*x^2+945*x^3-1152*x*z^2+1080*x^2-1152*x+128)*z^2/x^2/(3*x*z^2+3* x+4)/(27*z^4*x^3+27*z^2*x^3-27*z^2*x^2+1)/(x*z^2+x-1) ; # That operator is a symmetric product over the following extension: ( (1-x-x*z^2)*(z^4*x^3+z^2*x^3-z^2*x^2+1/27) )^(1/2); # which is still of genus 1. ######################################################################################### _Envdiffopdomain := [Dz,z]; Lz := Dz^4+(-896*y^6*z^5+704*y^5*z^6-68*y^4*z^7+1152*y^6*z^3-448*y^5*z^4+980*y^4*z^5-604*y^3*z^6-62*y^2*z^7+11*y*z^8-576*y^5*z^2-1196*y^4*z^3+392* y^3*z^4-238*y^2*z^5+126*y*z^6+36*z^7+60*y^4*z+484*y^3*z^2+402*y^2*z^3-98*y*z^4+2*y^2*z-98*y*z^2-48*z^3-5*y-12*z)/z/(z-1)/(z+1)/(-8*y^2*z+y*z ^2+y+3*z)/(-2*y*z+z^2+1)/(-8*y^3*z+6*y*z+z^2+1)*Dz^3+(-1280*y^6*z^5+1496*y^5*z^6-134*y^4*z^7+2432*y^6*z^3-2016*y^5*z^4+1438*y^4*z^5-1295*y^3 *z^6-232*y^2*z^7+31*y*z^8-920*y^5*z^2-2410*y^4*z^3+1764*y^3*z^4-97*y^2*z^5+272*y*z^6+111*z^7+98*y^4*z+755*y^3*z^2+835*y^2*z^3-441*y*z^4-96*z ^5-38*y^2*z-146*y*z^2-120*z^3-4*y-3*z)/z^2/(z-1)/(z+1)/(-8*y^2*z+y*z^2+y+3*z)/(-2*y*z+z^2+1)/(-8*y^3*z+6*y*z+z^2+1)*Dz^2+(-224*y^6*z^4+704*y ^5*z^5-53*y^4*z^6+1056*y^6*z^2-1648*y^5*z^3+281*y^4*z^4-614*y^3*z^5-214*y^2*z^6+22*y*z^7-288*y^5*z-887*y^4*z^2+1432*y^3*z^3+316*y^2*z^4+126* y*z^5+87*z^6+27*y^4+222*y^3*z+260*y^2*z^2-366*y*z^3-141*z^4-18*y^2-38*y*z-45*z^2+3)/z^2/(z-1)/(z+1)/(-8*y^2*z+y*z^2+y+3*z)/(-2*y*z+z^2+1)/(-\ 8*y^3*z+6*y*z+z^2+1)*Dz+(8*y^6*z^3+25*y^5*z^4-y^4*z^5+24*y^6*z-152*y^5*z^2+7*y^4*z^3-22*y^3*z^4-26*y^2*z^5+2*y*z^6-9*y^5+14*y^4*z+128*y^3*z^2 +56*y^2*z^3+3*y*z^4+9*z^5+6*y^3-14*y^2*z-36*y*z^2-21*z^3-y)/z^2/(z-1)/(z+1)/(-8*y^2*z+y*z^2+y+3*z)/(-2*y*z+z^2+1)/(-8*y^3*z+6*y*z+z^2+1) ; # Is a pullback of a smaller operator under this pullback: x = z/(z^2+1) namely: Lz_reduced_x := Dx^4+(7168*y^6*x^5-8064*y^4*x^5-3904*y^5*x^4-1152*y^6*x^3+2016*y^2*x^5+3296*y^3*x^4+1696*y^4*x^3+576*y^5*x^2-552*y*x^4-236*y^2*x^3-484*y^3*x^2-60*y^4*x-108*x^3 +38*y*x^2-2*y^2*x+12*x+5*y)/x/(2*x+1)/(2*x-1)/(8*y^2*x-3*x-y)/(2*y*x-1)/(8*y^3*x-6*y*x-1)*Dx^3+(26880*y^6*x^5-30240*y^4*x^5-12480*y^5*x^4-2432*y^6*x^3+7560*y^2 *x^5+10440*y^3*x^4+4008*y^4*x^3+920*y^5*x^2-1620*y*x^4-711*y^2*x^3-755*y^3*x^2-98*y^4*x-225*x^3+3*y*x^2+38*y^2*x+3*x+4*y)/x^2/(2*x+1)/(2*x-1)/(8*y^2*x-3*x-y)/( 2*y*x-1)/(8*y^3*x-6*y*x-1)*Dx^2+3*(8960*y^6*x^4-10080*y^4*x^4-3440*y^5*x^3-352*y^6*x^2+2520*y^2*x^4+2840*y^3*x^3+728*y^4*x^2+96*x*y^5-390*y*x^3-176*y^2*x^2-74* y^3*x-9*y^4-30*x^2-13*y*x+6*y^2-1)/x^2/(8*y^3*x-6*y*x-1)/(2*y*x-1)/(8*y^2*x-3*x-y)/(2*x+1)/(2*x-1)*Dx+3*(1120*y^5*x^3-1260*y^3*x^3-340*y^4*x^2-8*x*y^5+315*y*x^ 3+275*y^2*x^2+37*y^3*x+3*y^4-30*x^2-14*y*x-2*y^2-1)*y/x^2/(8*y^3*x-6*y*x-1)/(2*y*x-1)/(8*y^2*x-3*x-y)/(2*x+1)/(2*x-1); # That operator is a symmetric product over the following extension: ( -3*(x*y-1/2)*(x-1/2)*(x*y^3-3/4*x*y-1/8)*(x+1/2) )^(1/2); # which is still of genus 1. ##################### Theorem Legendre cube, series check ########################### H := p^(-1/2)*hypergeom([1/4, 3/4],[1],(y^2-1)^3*(x^2-1/4 * x^4)/p^2); p := (1-y^3*x+(3/4*y^2-1/2)*x^2); Legendre_cube := Sum(LegendreP(n,y)^3 * z^n, n=0..infinity); Theorem_Lcube := 1/sqrt(1+z^2) * Eval( Hadamard(H, 1/sqrt(1-4*x)), x=z/(1+z^2) ); ACC := 10; # Check series to accuracy 10 (i.e. mod z^10). value(eval(Theorem_Lcube, Hadamard = proc(f,g) global x, ACC; local f1,g1,n; f1, g1 := series(f,x=0,ACC), series(g,x=0,ACC); add( coeff(f1,x,n)*coeff(g1,x,n)*x^n, n=0..ACC-1) end )): ShouldBeZero := simplify(series(% - Legendre_cube, z=0, ACC));