Superimposing theta structure on a generalized modular relation

Abstract

A generalized modular relation of the form $F(z, w, \alpha)=F(z, iw,\beta)$, where $\alpha\beta=1$ and $i=\sqrt{-1}$, is obtained in the course of evaluating an integral involving the Riemann $\Xi$-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 $\zeta(s, a)$, which we denote by $\zeta_w(s, a)$. While $\zeta_w(s, 1)$ is essentially a product of confluent hypergeometric function and the Riemann zeta function, $\zeta_w(s, a)$ for $0<a<1$ is an interesting new special function. We show that $\zeta_w(s, a)$ satisfies a beautiful theory generalizing that of $\zeta(s, a)$ albeit the properties of $\zeta_w(s, a)$ are much harder to derive than those of $\zeta(s, a)$. In particular, it is shown that for $0<a<1$ and $w\in\mathbb{C}$, $\zeta_w(s, a)$ 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 $\zeta_w(s, a)$. The theory of functions reciprocal in the kernel $\sin(\pi z) J_{2 z}(2 \sqrt{xt}) -\cos(\pi z) L_{2 z}(2 \sqrt{xt})$, where $L_{z}(x)=-\frac{2}{\pi}K_{z}(x)-Y_{z}(x)$ and $J_{z}(x), Y_{z}(x)$ and $K_{z}(x)$ are the Bessel functions, is worked out. So is the theory of a new generalization of $K_{z}(x)$, namely, ${}_1K_{z,w}(x)$. Both these theories as well as that of $\zeta_w(s, a)$ are essential to obtain the generalized modular relation.