q-Analogues of several $\pi$-formulas

TL;DR

This paper develops $q$-analogues of several classical $ ext{pi}$-formulas using $q$-series and telescoping methods, providing new proofs and extensions of known identities.

Contribution

It introduces new $q$-analogues of $ ext{pi}$-formulas and offers a short proof of Hou and Sun's identity using $q$-series techniques.

Findings

Established $q$-analogues of multiple $ ext{pi}$-formulas

Provided a concise proof of Hou and Sun's identity

Extended known $ ext{pi}$-formulas to the $q$-series context

Abstract

According to the $q$-series method, a short proof for Hou and Sun's identity, which is the $q$-analogue of a known $\pi$-formula, is offered. Furthermore, $q$-analogues of several other $\pi$-formulas are also established in terms of the $q$-series method and the telescoping method.