Transformation and summation formulas for basic hypergeometric series associated with the circumference ratio

TL;DR

This paper develops new transformation and summation formulas for basic hypergeometric series, leading to novel q-analogues of famous mathematical series related to pi and zeta functions.

Contribution

It introduces several new formulas connecting hypergeometric series with q-analogues of classical pi and zeta series, expanding the understanding of these special functions.

Findings

q-analogues of Ramanujan's series for 1/π derived

New q-series for ζ(3) established

Transformation formulas enhance hypergeometric series theory

Abstract

In terms of the difference operators, we establish several curious transformation and summation formulas for basic hypergeometric series. When the parameters are specified, they produce $q$-analogues of Ramanujan's three series for 1/$\pi$ and other eleven nice $\pi$-formulas. Meanwhile, $q$-analogues of three beautiful series for $\zeta(3)$ are also given in the same way.