# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1904.01767