Ramanujan-type formulae for $1/\pi$: $q$-analogues

TL;DR

This paper explores $q$-analogues of Ramanujan's hypergeometric formulas for $1/\pi$, focusing on their arithmetic properties and supercongruences, advancing understanding of these classical formulas in a modern $q$-series context.

Contribution

It introduces new $q$-analogues of Ramanujan-type formulas for $1/\pi$ and investigates their supercongruences, extending classical results into the realm of $q$-series.

Findings

Derived new $q$-analogues of Ramanujan's formulas for $1/\pi$

Established supercongruences related to these $q$-analogues

Enhanced understanding of the arithmetic properties of Ramanujan-type formulas

Abstract

The hypergeometric formulae designed by Ramanujan more than a century ago for efficient approximation of $\pi$, Archimedes' constant, remain an attractive object of arithmetic study. In this note we discuss some $q$-analogues of Ramanujan-type evaluations and of related supercongruences.