Proving some identities of Gosper on $q$-trigonometric functions

TL;DR

This paper uses elliptic function theory to prove three identities involving Gosper's $q$-trigonometric functions, which serve as $q$-analogues of classical sine and cosine functions.

Contribution

It provides the first rigorous proofs of several of Gosper's conjectured identities for $q$-trigonometric functions using elliptic functions.

Findings

Proved three identities of Gosper involving $q$-sine and $q$-cosine.

Established connections between $q$-trigonometric functions and elliptic functions.

Enhanced understanding of $q$-analogues of classical trigonometric identities.

Abstract

Gosper introduced the functions $\sin_q z$ and $\cos_q z$ as $q$-analogues for the trigonometric functions $\sin z$ and $\cos z$ respectively. He stated but did not prove a variety of identities involving these two $q$-trigonometric functions. In this paper, we shall use the theory of elliptic functions to prove three formulas from the list of Gosper on the functions $\sin_q z$ and $\cos_q z$.