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arXiv · 2504.15161

A few identities and integrals involving Pochhammer symbols, Jacobi polynomials, and the generalized hypergeometric function

Paweł J. Szabłowski
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Abstract

We first present some identities involving the Pochhammer symbol (rising factorial). We also recall and present some new properties of the Jacobi polynomials. We use them to expand a general hypergeometric function in an orthogonal series of Jacobi polynomials. Then we use these expansions to discover closed forms for certain integrals of Jacobi polynomials that are multiplied by a generalized hypergeometric function and a Beta density. We can also obtain closed forms for some series involving rising factorials that generalise binomial series by using well-known properties of the hypergeometric function. In particular, we get a few new, nontrivial identities involving the Pochhammer symbol. We can also derive some simplifying identities for generalized hypergeometric functions.

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Paweł J. Szabłowski. 2025-04-21. A few identities and integrals involving Pochhammer symbols, Jacobi polynomials, and the generalized hypergeometric function. https://arxiv.org/abs/2504.15161

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