A note on an integral of Dixit, Roy and Zaharescu

TL;DR

This paper investigates a specific integral related to a recent open question, expressing it in terms of hypergeometric functions and gamma functions, and provides bounds on the remainder term showing exponential decay for large parameters.

Contribution

It derives a closed-form expression for the integral involving hypergeometric and gamma functions, addressing an open question from prior research.

Findings

Integral expressed in terms of hypergeometric and gamma functions

Remainder term bounded and shown to decay exponentially for large k

Provides partial closed-form evaluation for the integral

Abstract

In a recent paper, Dixit {\it et al.\/} [Acta Arith. {\bf 177} (2017) 1--37] posed two open questions whether the integral \[{\hat J}_{k}(\alpha)=\int_0^\infty\frac{xe^{-\alpha x^2}}{e^{2\pi x}-1}\,{}_1F_1(-k,3/2;2\alpha x^2)\,dx\] for $\alpha>0$ could be evaluated in closed form when $k$ is a positive even and odd integer. We establish that ${\hat J}_{k}(\alpha)$ can be expressed in terms of a Gauss hypergeometric function and a ratio of two gamma functions, together with a remainder expressed as an integral. An upper bound on the remainder term is obtained, which is shown to be exponentially small as $k$ becomes large when $a=O(1)$.