newton1
Nov29-03, 07:37 AM
i need to prove generating function G(u,x)=1/(1-2xu+u^2)^0.5
G(u,x)=\sum (n->infinite) Pn(x) u^n , where Pn is legendre polynimial Pn=\sum (m->imfinite) [(1)^m(2n-2m)!x^(n-2m)/(2^n)m!(n-m)!(n-2m)!]
let v= 2xu+u^2
1/(1-v)^0.5 = 1 + [1/2] v + [(1*3)/(2^2 * 2!)] v^2+....+ [1*3*5*...*(2n-1)/2^n * n!]v^n + .....
so,
1/(1-2xu+u^2)^0.5= 1 + [1/2] u(2x-u) + [(1*3)/2^2 * 2!)] u^2(2x-u)^2+....+ [1*3*5*...*(2n-1)/2^n * n!]u^n(2x-u)^n + .....
and i don't know how to deduce to the legendre polynomial
G(u,x)=\sum (n->infinite) Pn(x) u^n , where Pn is legendre polynimial Pn=\sum (m->imfinite) [(1)^m(2n-2m)!x^(n-2m)/(2^n)m!(n-m)!(n-2m)!]
let v= 2xu+u^2
1/(1-v)^0.5 = 1 + [1/2] v + [(1*3)/(2^2 * 2!)] v^2+....+ [1*3*5*...*(2n-1)/2^n * n!]v^n + .....
so,
1/(1-2xu+u^2)^0.5= 1 + [1/2] u(2x-u) + [(1*3)/2^2 * 2!)] u^2(2x-u)^2+....+ [1*3*5*...*(2n-1)/2^n * n!]u^n(2x-u)^n + .....
and i don't know how to deduce to the legendre polynomial