Modular Proofs of Gosper's Identities

TL;DR

This paper provides unified modular proofs for all of Gosper's identities on the $q$-constant $ ext{Pi}_q$, confirms polynomial relations among these constants, and introduces a method to rediscover and construct such identities.

Contribution

It offers a unified modular proof framework for Gosper's identities, confirms polynomial relations among $ ext{Pi}_q$ constants, and presents a strategy using hauptmoduls for constructing identities.

Findings

Unified modular proofs for all Gosper's identities on $ ext{Pi}_q$.

Confirmation that $ ext{Pi}_{q^{n_i}}$ satisfy nonzero polynomial relations for distinct $n_i$.

Correction of several results on $ ext{Pi}_q$ by El Bachraoui.

Abstract

We give unified modular proofs to all of Gosper's identities on the $q$-constant $\Pi_q$. We also confirm Gosper's observation that for any distinct positive integers $n_1,\cdots,n_m$ with $m\geq 3$, $\Pi_{q^{n_1}}$, $\cdots$, $\Pi_{q^{n_m}}$ satisfy a nonzero homogeneous polynomial. Our proofs provide a method to rediscover Gosper's identities. Meanwhile, several results on $\Pi_q$ found by El Bachraoui have been corrected. Furthermore, we illustrate a strategy to construct some of Gosper's identities using hauptmoduls for genus zero congruence subgroups.