Proofs for certain Pi_{q}-conjectures of Gosper

TL;DR

This paper proves certain conjectured identities involving the Pi_{q} constant introduced by Gosper, using modular equations of degrees 3 and 5, thereby confirming some of Gosper's unproven conjectures.

Contribution

The paper establishes modular equations of degrees 3 and 5 and uses them to prove specific Pi_{q}-identities conjectured by Gosper.

Findings

Confirmed two groups of Pi_{q}-identities in Gosper's list.

Established modular equations of degrees 3 and 5.

Validated conjectures involving Pi_{q} and its related constants.

Abstract

In 2001 W. Gosper introduced a constant Pi_{q} and conjectured without proofs many intriguing identities on this constant. In this paper we establish some modular equations of degrees 3 and 5. From these modular equations we confirm two groups of Pi_{q}-identities in Gosper's list. One group involves Pi_{q},Pi_{q^{2}},Pi_{q^{3}} or Pi_{q^{6}} while the other is related to Pi_{q},Pi_{q^{2}},Pi_{q^{5}} or Pi_{q^{10}}.