A few finite and infinite identities involving Pochhammer and $q$- Pochhammer symbols obtained via analytical methods
Paweł J. Szabłowski
TL;DR
The paper develops a unified analytic framework to derive identities involving Pochhammer and q-Pochhammer symbols by expanding the Radon–Nikodym derivative between two orthogonality measures in the connection-coefficient basis. Using Jacobi polynomials and the Beta distribution, it translates polynomial recurrences into explicit two-variable identities for Pochhammer symbols; using the Askey–Wilson scheme it does the same for q-Pochhammer symbols across eight suggested polynomial pairings. The main contributions are eight concrete finite/ infinite identities linking $q$-Pochhammer objects and classical Pochhammer structures, obtained by density-expansion arguments and coefficient-relationship in the CC matrices. The results provide a coherent method to generate and justify a large class of identities, with potential applications in combinatorics, hypergeometric transformations, and $q$-series transformations. Overall, the work offers a principled route to simplify calculations involving Pochhammer symbols and to discover new algebraic relations among orthogonal-polynomial families.
Abstract
We present several identities with a form of polynomials or rational functions that involve Pochhammer and q-Pochhammer symbols and q-binomials (i.e. Gauss polynomials). All these identities were obtained by some analytical methods based on infinite expansions of the ratio of densities in a Fourier series of polynomials orthogonal with respect to the density in the denominator. We want a unified approach to justify many known and unknown identities. The purpose of studying these identities is to simplify calculations occurring while dealing with Pochhammer and q-Pochhammer symbols. Additional possible applications of the results presented in the paper are applications within the Combinatorics and the transformation formulae of hypergeometric and basic hypergeometric functions.