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The following integral came up in one of my studies:

$$\int_{-1}^1 (1-x)^\alpha (1+x)^\beta P_n^{(\alpha,\beta)}(x)\,P_{n+j}(x)\,dx$$

where $P_n^{(\alpha,\beta)}(x)$ is a Jacobi polynomial and $P_m(x)$ is a Legendre polynomial. I would like to know if this integral has a closed form. As usual, $\alpha>-1$ and $\beta>-1$, and I assume $n$ is a nonnegative integer, and $j$ is a positive integer.

Experiments with evaluating this integral with rational values of $\alpha,\beta$ have led me to suspect that this should have a closed form, possibly involving gamma functions or hypergeometric functions. Unfortunately, my best efforts to search through Abramowitz and Stegun, Gradshteyn and Ryzhik, the DLMF, and the Wolfram Functions Site have been to no avail.

Alternatively, if it should somehow be simpler, I would also be interested in seeing the closed form for

$$\int_{-1}^1 (1-x)^\alpha (1+x)^\beta P_n^{(\alpha,\beta)}(x)\,x^{n+j}\,dx$$

with the same parameter conditions as above, since I can convert the result of this to the result of the first integral, thanks to the orthogonality property of the Jacobi polynomial.

References to where integrals of this type have been previously discussed are very much welcome as well, and will be quite appreciated.

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I don't have time to work out the details, and I don't know know if the OP would accept a double sum as a 'closed form' answer, but this how one can obtain such an expression.

The Legendre polynomial can be written in terms as a Jacobi polynomial as $P_n(x)=P_n^{(0,0)}(x).$ Examine DLMF 18.18.14 and you'll see and expression of the form

$$ P_n^{(c,b)}(x)=\sum_{k=0}^m A_k(a,b,c)P_k^{(a,b)}(x) $$ where the $A_k$ are ratios of pochhammer (gamma function) symbols. Note that the 'b' parameter is fixed. Set $c=b=0.$ Then the inner Jacobi poly becomes $P_k^{(a,0)}(x).$ Use the Jacobi symmetry $P_k^{(a,b)}(-x)=(-1)^k P_k^{(b,a)}(x)$ to switch the order of the indices. Then you can use the previous formula again and with some relabeling you can get the innermost Jacobi polynomial to have the form $P_L^{(\alpha,\beta)}(x).$ Once in this form your integral can be performed analytically by orthogonality of the Jacobi polynomials. I doubt the double sum simplifies.

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