A few identities and integrals involving Pochhammer symbols, Jacobi polynomials, and the generalized hypergeometric function
Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Szabłowski, Paweł J.
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416872440