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 \[{\hat J}_{k}(α)=\int_0^\infty\frac{xe^{-αx^2}}{e^{2πx}-1}\,{}_1F_1(-k,3/2;2αx^2)\,dx\] 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 $a=O(1)$.

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