Some new sums of $q$-trigonometric and related functions through a theta product of Jacobi

TL;DR

This paper derives new finite and infinite sums involving q-trigonometric and q-digamma functions using Jacobi's theta product formula, connecting q-analogues to classical functions as q approaches 1.

Contribution

It introduces novel sums involving q-trigonometric and q-digamma functions derived from theta product identities, bridging q-analogues with classical functions.

Findings

Derived new sums involving q-trigonometric functions.

Connected q-analogues to classical functions as q approaches 1.

Utilized theta product formula of Jacobi and Gosper's identities.

Abstract

We evaluate some finite and infinite sums involving $q$-trigonometric and $q$-digamma functions. Upon letting $q$ approach $1$, one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a theta product formula of Jacobi and Gosper's $q$-trigonometric identities.