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

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 $w\to 0$ 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.