A note on an integral of Dixit, Roy and Zaharescu

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 $a=O(1)$.