Hostname: page-component-7c8c6479df-27gpq Total loading time: 0 Render date: 2024-03-28T16:42:13.746Z Has data issue: false hasContentIssue false

Analogues of the general theta transformation formula

Published online by Cambridge University Press:  18 March 2013

Atul Dixit*
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA

Abstract

A new class of integrals involving the confluent hypergeometric function 1F1(a;c;z) and the Riemann Ξ-function is considered. It generalizes a class containing some integrals of Ramanujan, Hardy and Ferrar and gives, as by-products, transformation formulae of the form F(z, α) = F(iz, β), where αβ = 1. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulae 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 Littlewood involving infinite series of the Möbius function.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)