Analogues of the general theta transformation formula

Abstract

A new class of integrals involving the confluent hypergeometric function ${}_1{F}_1(a;c;z)$ 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 Möbius functions.

27 pages