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Integration of hypergeometric product for legendre polynomials

I'm looking for a general solution to the integral: $\int_{0}^1 {_2}F_1(m-n,m+n+1,m+1;z){_2}F_1(m-k+1,m+k+2,m+2;z) dz$ where $m,n,k\in \mathbb{N}$ and $m\leqslant n$ and $m+1 \leqslant k$. To give …
Will Trojak's user avatar