Mathematica Eterna
Open Access

Abstract

A note on an integral of Dixit, Roy and Zaharescu

R. B. Paris

In a recent paper, Dixit et al. [Acta Arith. 177 (2017) 1–37] posed two open questions whether the integral Jˆ k(α) = Z ∞ 0 xe−αx2 e 2πx − 1 1F1(−k, 3 2 ; 2αx2 ) dx for α > 0 could be evaluated in closed form when k is a positive even and odd integer. We establish that Jˆ k(α) 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).