Proofs for certain conjectures of Gosper on q-trigonometric identities

TL;DR

This paper uses elliptic functions to prove two conjectured q-trigonometric identities by Gosper, providing new proofs and establishing key theta function identities.

Contribution

It introduces a novel elliptic function approach to confirm Gosper's conjectures and simplifies the proof of a Pi_q-identity.

Findings

Confirmed two Gosper conjectured q-trigonometric identities

Established two new Jacobi theta function identities

Provided a simplified proof of a Pi_q-identity

Abstract

Applying the theory of elliptic functions we establish two Jacobi theta function identities. From these identities we confirm two q-trigonometric identities conjectured by Gosper. As an application, we give a new and simple proof of a Pi_{q}-identity of Gosper.