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

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.

💡 Research Summary

The paper investigates a collection of identities involving the rising factorial (Pochhammer symbol) and exploits these identities to develop orthogonal expansions of generalized hypergeometric functions in terms of Jacobi polynomials. The authors begin by recalling the definition ((x)_n=\Gamma(x+n)/\Gamma(x)) and several elementary properties, such as the relation between rising and falling factorials and the Chu‑Vandermonde summation formula. Lemma 1 presents three non‑trivial summation formulas (equations (2.1)–(2.4)) that express reciprocals of Pochhammer symbols and certain double sums as Kronecker deltas.

Next, the paper introduces the standard Jacobi polynomials (J_n(x\mid a,b)) and a closely related family (K_n(x\mid a,b)) obtained by a simple linear change of variable. Explicit coefficient functions (e_{n,m}(a,b)) and (\tilde e_{n,m}(a,b)) are defined so that (J_n) and (K_n) can be written as finite linear combinations of powers ((x-1)^m) and vice‑versa (equations (2.12)–(2.13)). The authors also list symmetry relations (2.16)–(2.18) that connect the two families under sign changes and parameter swaps.

A central part of the work is Proposition 1, which evaluates integrals of the form (\int_0^1 K_n(x\mid a,c)f(x\mid a,b),dx) and (\int_0^1 K_n(x\mid c,b)f(x\mid a,b),dx), where (f) denotes the Beta density. The results are compact expressions involving products of Pochhammer symbols: \