Series involving product of Legendre polynomials

Question

I need to compute the following sum: $$\sum_{n=0}^{\infty} (4n+3) P_{2n+1}(x)P_{2n+1}(y)$$ where $P_n(x)$ are the Legendre polynomials. Can anyone help me?

Answer 1

The Legendre polynomials satisfy the orthogonality relation

$$ \int_{-1}^1 P_m(x)P_n(x)\,\mathrm{d}x=\frac{2}{2n+1}\delta_{mn}. $$

Therefore, if we apply the kernel

$$ K(x,y)=\sum_{m=0}^\infty (4m+3)P_{2m+1}(x)P_{2m+1}(y) $$

as an integral transform of a function decomposed as $f(y)= \displaystyle \sum_{n=0}^\infty f_n P_n(y)$ we get

$$ \int_{-1}^1 K(x,y)f(y)\,\mathrm{d}y=\sum_{m,n=0}^\infty (4m+3)P_{2m+1}(x) \left[ \int_{-1}^1 {\color{red}{P_{2m+1}(y)}}P_{n}(y)\,\mathrm{d}y \right] f_n $$

$$ =\sum_{m=0}^\infty (4m+3)P_{2m+1}(x) \frac{2}{2(2m+1)+1}f_{2m+1}=2\sum_{m=0}^\infty P_{2m+1}(x)f_{2m+1} $$

$$ = f(x)-f(-x)=\int_{-1}^1 \big[\delta(y-x)-\delta(y+x)\big]f(y)\,\mathrm{d}y. $$

Thus, the kernel function is $K(x,y)=\delta(x-y)-\delta(x+y)$.