A Ramanujan's hypergeometric transformation formula, its validity range and implications

TL;DR

This paper extends the validity of a Ramanujan hypergeometric transformation, explores its implications for elliptic integrals, and derives new closed-form evaluations of hypergeometric functions.

Contribution

It broadens the known validity range of a key hypergeometric transformation and links it to elliptic integrals and other transformations.

Findings

Extended the validity range of Ramanujan's hypergeometric transformation.

Derived new closed-form evaluations of $_2F_1$ hypergeometric functions.

Established connections with elliptic integrals and recent hypergeometric results.

Abstract

We extend the validity range of a Ramanujan's hypergeometric transformation formula proved by Berndt, Bhargava and Garvan, Trans. Amer. Math. Soc. 347, 4163 (1995) and study its implications. Relations to special values of complete elliptic integrals of the first kind in the singular value theory are established. Consequently, we derive several closed-form evaluations of hypergeometric functions $_2F_1$ with different sets of parameters and arguments. Connections with other hypergeometric transformations and some recent results are discussed.