A note on an integral of Dixit, Roy and Zaharescu

2018/04/20 by R. B. Paris, Paris, R B
Mathematics · #30E20 #33C05 #33C15 #34E05 #41A60 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1804.07527

openalex publication_date 2018/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a recent paper, Dixit \it et al.\/ [Acta Arith. \bf 177 (2017) 1--37] posed two open questions whether the integral J_k(α)=∫₀^∞\fracxe^-αx2e^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 J_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).

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