Analogues of the general theta transformation formula

作者：Dixit Atul^*

来源：Proceedings of the Royal Society of Edinburgh: Section A Mathematics , 2013, 143(2): 371-399.

DOI：10.1017/S0308210511001685

A new class of integrals involving the confluent hypergeometric function F-1(1)(a;c;z) and the Riemann Xi-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, 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 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 Mobius function.

• 出版日期2013-4