A note on an integral of Dixit, Roy and Zaharescu

In a recent paper, Dixit {\it et al./} [Acta Arith. {\bf 177} (2017) 1–37] posed two open questions whether the integral $${\widehat{J}}_k(\alpha )={\int }_0^{\mathrm{\infty }}\frac{x{e}^{-\alpha x^2}}{e^{2\pi x}-1}{}_1{F}_1(-k,3\mathrm{/}2;2\alpha x^2)dx$$ for $\alpha >0$ could be evaluated in closed form when $k$ is a positive even and odd integer. We establish that ${\widehat{J}}_k(\alpha )$ 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 $a=O(1)$.