A few finite and infinite identities involving Pochhammer and $q$- Pochhammer symbols obtained via analytical methods

TL;DR

This paper derives new finite and infinite identities involving Pochhammer and q-Pochhammer symbols using analytical methods, aiming to simplify calculations and facilitate applications in combinatorics and hypergeometric functions.

Contribution

It introduces a unified analytical approach to derive and justify various identities involving Pochhammer and q-Pochhammer symbols, both known and novel.

Findings

Derived several identities involving Pochhammer and q-Pochhammer symbols

Provided a unified analytical framework for these identities

Potential applications in combinatorics and hypergeometric transformations

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.