Using generating function for Legendre polynomials, prove. sum_0^l(2l+1)P_l(x)t^l=(1-t^2)/((1-2xt+t^2)^(3/2))

aflacatn

aflacatn

Answered question

2021-09-04

Using generating function for Legendre polynomials, prove that
0l(2l+1)Pl(x)tl=1t2(12xt+t2)32

Answer & Explanation

jlo2niT

jlo2niT

Skilled2021-09-05Added 96 answers

Introduction
The legendre polynomial denoted by Pn(x).
The solution of physical problems is the class of function called legendre polynomials.
Explanation
To prove,
0lPl(x)tl(l+l+1)=xtt2+xtt2(12xt+t2)32+1(12xt+t2)12
Use generating function for legendre polynomial,
x=0Pn(x)tn=(12xt+t2)12
Consider the limit 0 to l
n=0lPl(x)tl=(12xt+t2)12
Differentiating with respect to t,
0lPl(x)<l1=(12)(12xt+t2)32(2x+2t)
0lPl(x)<l1=(xt)(12xt+t2)32
Multiplying t on both sides we get,
0lPl(x)<l1t=(xt)t(12xt+t2)32
0lPl(x)<l=xtt2(12xt+t2)32
Let us consider,

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