[math-fun] discriminant of Legendre polynomials

Greg Fee wrote [>PS, John Brillhart challenged me to find the discriminant of the nth Legendre [>polynomial discriminant. I hope he doesn't mind my ducking and throwing open [>the question... The discriminants of many orthogonal polynomials can be found in the book "Orthogonal Polynomials" by Gabor Szego. (section 6.7) For the Legendre polynomials, if I've copied it right, you get 2^{-n(n-1)} product_{i=1}^n i^{3i-2n} (n+i)^{n-i}. One proves this using the formula for discriminants involving the derivative: D(f)=(-1)^{n(n-1)/2} a^{n-2} product_{i=1}^n f ' (alpha_i) where a=leading coefficient of f, alpha_i are the roots of f, f ' =derivative of f. Then use the differential equation to get f ' (alpha_i). Gary McGuire