A modular relation involving a generalized digamma function and asymptotics of some integrals containing $\Xi(t)$

TL;DR

This paper establishes a new modular relation involving a generalized digamma function, explores its connection to Ramanujan's classical results, and derives asymptotic estimates for integrals involving the Riemann $\Xi(t)$ function.

Contribution

It introduces a novel modular relation with the generalized digamma function and extends Ramanujan's classical results to a broader context.

Findings

Derived a modular relation involving the generalized digamma function.

Obtained asymptotic estimates for integrals with the Riemann $\Xi(t)$ function.

Connected the new relation to Ramanujan's classical results.

Abstract

A modular relation of the form $F(\alpha, w)=F(\beta, iw)$, where $i=\sqrt{-1}$ and $\alpha\beta=1$, is obtained. It involves the generalized digamma function $\psi_w(a)$ which was recently studied by the authors in their work on developing the theory of the generalized Hurwitz zeta function $\zeta_w(s, a)$. The limiting case $w\to0$ of this modular relation is a famous result of Ramanujan on page $220$ of the Lost Notebook. We also obtain asymptotic estimate of a general integral involving the Riemann function $\Xi(t)$ as $\alpha\to\infty$. Not only does it give the asymptotic estimate of the integral occurring in our modular relation as a corollary but also some known results.