I have a couple of integrals of triple infinite sums of associated Legendre polynomials, which I'd like to integrate using Gaunt's Formula. Any help would be very much appreciated, as I'm really confused by when the rules Wikipedia supplies for when Gaunt's formula can be applied. So the integrals are of the form: $$\int_{-1}^1 \sum_{m,n,k=0}^\infty P_k^{(i)}(x)P_m^{(1)}(x) P_n^{(1)}(x) dx $$ for $i=0$ and $i=2$ where $P_\ell^{(u)}$ is the associated Legendre polynomial.
So working with the $i=2$ case Gaunt's formula tells us:
$$\frac{1}{2}\int_{-1}^1 P_k^{(2)}(x)P_m^{(1)}(x) P_n^{(1)}(x) dx = (-1)^{s-m-1}\frac{(m+1)!(n+1)!(2s-2n)!s!}{(m-1)!(s-k)!(s-m)!(s-n)!(2s+1)!}\times \ \sum_{t=0}^q (-1)^t \frac{(k+u+t)!(m+n-2-t)!}{t!(k-2-t)!(m-n+2+t)!(n-1-t)!} $$ where $2s=k+m+n$ and $q=\min( m+n-2,k-2,n-1)$.
This formula can be used when:
- Degrees (k,m,n) and orders of the Legendre-polynomials are non negative integers (tick)
- The first of the three orders is the largest (tick)
- The orders sum up (2=1+1: tick)
- The degrees obey $m\ge n$ (not sure how this affects the sum....)
However the integral equals $0$ unless:
- $s$ is an integer, i.e. $k+m+n$ is even
- $m+n\ge k \ge m-n$
So the part I'm confused with is how the rules affecting when the integral is zero, and when Gaunt's formula can be applied, affect what elements need to be summed over, as it seems that $$\sum_{m,n,k=0}^\infty\frac{1}{2}\int_{-1}^1 P_k^{(2)}(x)P_m^{(1)}(x) P_n^{(1)}(x) dx \ne\sum_{m,n,k=0}^\infty (-1)^{s-m-1}\frac{(m+1)!(n+1)!(2s-2n)!s!}{(m-1)!(s-k)!(s-m)!(s-n)!(2s+1)!}\times \ \sum_{t=0}^q (-1)^t \frac{(k+u+t)!(m+n-2-t)!}{t!(k-2-t)!(m-n+2+t)!(n-1-t)!} $$
If it would be helpful for me to add any extra details please let me know as I'm new on here.