Special values of $q$-gamma products

TL;DR

This paper explores special values of products of $q$-gamma functions with rational arguments, extending classical gamma function results to the $q$-analogue and providing explicit evaluations involving $e^{-\pi}$ and gamma functions.

Contribution

It introduces new $q$-generalizations of gamma function product formulas, especially for products indexed by Dirichlet characters, and computes explicit values.

Findings

Derived new $q$-product identities involving Dirichlet characters.

Provided explicit evaluations of infinite $q$-gamma products in terms of $e^{-\pi}$ and gamma functions.

Extended classical gamma function product results to the $q$-analogue setting.

Abstract

We consider products of $q$-gamma functions with rational arguments, and prove several $q$-generalizations of recent works concerning products of gamma functions. In particular, we consider products indexed by Dirichlet characters, and provide several new values for infinite products in terms of $e^{-\pi}$ and gamma functions.