Formula_Pn_square := Sum( binomial(n,k) * binomial(n+k,n) * binomial(2*k,k) * ((y^2-1)/4)^k, k=0..n);
ShouldBeZero := simplify(value(eval(LegendreP(n,y)^2 - Formula_Pn_square, n=11))); # checked for random n

# Creative telescoping produced:
SystemSection2 := {
  z*(y^2-2*z-1/2)*diff(F(y,z), y,z) + y/2 * (1-y^2) * diff(F(y,z), y,y)
      - (y^2+z) * diff(F(y,z), y) + y*z*diff(F(y,z), z),
  F(y,z) + (2*z^2 - z/2) * diff(F(y,z), z,z) + (5*z - 1/2) * diff(F(y,z), z)
      + (y-1/y) * ((z+1/4) * diff(F(y,z), y,z) + 1/2 * diff(F(y,z), y)) }:

# Elimination produced DEz and DEy (too large to print in paper):
DEz := (4*z - 1)*z^2*(16*y^2*z - 16*z^2 - 8*z - 1)*(8*z - 1)*diff(F(z), z,z,z,z)
  + z*(4608*y^2*z^3 - 1472*y^2*z^2 - 5888*z^4 + 120*y^2*z - 352*z^3 + 368*z^2 + 2*z - 5)*diff(F(z), z,z,z)
  + (9728*y^2*z^3 - 2624*y^2*z^2 - 17280*z^4 + 196*y^2*z + 144*z^3 + 668*z^2 - 38*z - 4)*diff(F(z), z,z)
  + (4224*y^2*z^2 - 1008*y^2*z - 12480*z^3 + 54*y^2 + 1032*z^2 + 276*z - 18)*diff(F(z), z)
  + (96*y^2*z - 36*y^2 - 960*z^2 + 168*z + 12)*F(z);

DEy := (y - 1)^2*(8*y^2*z + y^2 + 8*z^2 - 6*z - 2)*(16*y^2*z - 16*z^2 - 8*z - 1)*(y + 1)^2*y*diff(F(y), y,y,y,y)
  + (y^2 - 1)*(1536*y^6*z^2 + 192*y^6*z + 896*y^4*z^3 - 2160*y^4*z^2 - 1152*y^2*z^4 - 592*y^4*z - 608*y^2*z^3
    - 7*y^4 + 792*y^2*z^2 - 128*z^4 + 278*y^2*z + 32*z^3 + 15*y^2 + 72*z^2 + 22*z + 2)*diff(F(y), y,y,y)
  + 2*(2368*y^6*z^2 + 296*y^6*z + 2816*y^4*z^3 - 3960*y^4*z^2 - 1216*y^2*z^4 - 1020*y^4*z - 2544*y^2*z^3 - 5*y^4
    + 1920*y^2*z^2 + 192*z^4 + 682*y^2*z - 16*z^3 + 15*y^2 - 136*z^2 - 38*z - 2)*y*diff(F(y), y,y)
  + (3712*y^6*z^2 + 464*y^6*z + 7040*y^4*z^3 - 4320*y^4*z^2 - 1024*y^2*z^4 - 1440*y^4*z - 2688*y^2*z^3
    - 2*y^4 + 1056*y^2*z^2 - 384*z^4 + 436*y^2*z + 32*z^3 + 6*y^2 + 272*z^2 + 76*z + 4)*diff(F(y), y)
  + 48*z*(8*y^2*z + y^2 + 24*z^2 - 6*z - 3)*y^3*F(y);

# Check that System_Section2, DEz and DEy are correct up to order 20:
ORD := 20;
Fs := add( binomial(2*n, n) * simplify(LegendreP(n,y))^2 * z^n, n=0..ORD+1):
ShouldBeZeros := seq(
	expand(series(subs(F(y) = Fs, F(z) = Fs, F(y,z) = Fs, i), z=0, ORD)),
	i in SystemSection2 union {DEz, DEy});
# Verified up to O(z^20).