Analogues of the general theta transformation formula

TL;DR

This paper introduces a new class of integrals involving hypergeometric and Riemann Xi functions, leading to generalized theta transformation formulas, extensions of Ramanujan's identities, and a broader conjecture involving M"obius series.

Contribution

It develops a novel class of integrals that generalize classical transformation formulas and identities related to theta functions and Ramanujan's work.

Findings

Derived an extended theta transformation formula.

Generalized Ramanujan's identity involving infinite series.

Proposed a new conjecture related to M"obius functions.

Abstract

A new class of integrals involving the confluent hypergeometric function ${}_1F_{1}(a;c;z)$ and the Riemann $\Xi$-function is considered. It generalizes a class containing some integrals of S. Ramanujan, G.H. Hardy and W.L. Ferrar and gives as by-products, transformation formulas of the form $F(z,\alpha)=F(iz,\beta)$, where $\alpha\beta=1$. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulas of Ferrar and Hardy. A one-variable generalization of a well-known identity of Ramanujan is also given. We conclude with a generalization of a conjecture due to Ramanujan, Hardy and J.E. Littlewood involving infinite series of M\"obius functions.