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

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.