Superimposing theta structure on a generalized modular relation

Abstract

A generalized modular relation of the form F(z, w, α) = F(z, iw, β), where αβ = 1 and i = √ −1, is obtained in the course of evaluating an integral involving the Riemann Ξ-function. It is a two-variable generalization of a transformation found on page 220 of Ramanujan’s Lost Notebook. This modular relation involves a surprising generalization of the Hurwitz zeta function ζ(s, a), which we denote by ζw(s, a). While ζw(s, 1) is essentially a product of confluent hypergeometric function and the Riemann zeta function, ζw(s, a) for 0 < a < 1 is an interesting new special function. We show that ζw(s, a) satisfies a beautiful theory generalizing that of ζ(s, a) albeit the properties of ζw(s, a) are much harder to derive than those of ζ(s, a). In particular, it is shown that for 0 < a < 1 and w ∈ C, ζw(s, a) can be analytically continued to Re(s) > −1 except for a simple pole at s = 1. This is done by obtaining a generalization of Hermite’s formula in the context of ζw(s, a). The theory of functions reciprocal in the kernel sin(πz)J2z(2√ xt) − cos(πz)L2z(2√ xt), where Lz(x) = − 2 π Kz(x) − Yz(x) and Jz(x), Yz(x) and Kz(x) are the Bessel functions, is worked out. So is the theory of a new generalization of Kz(x), namely, 1Kz,w(x). Both these theories as well as that of ζw(s, a) are essential to obtain the generalized modular relation.