English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In a recent paper, Dixit {\it et al.\/} [Acta Arith. {\bf 177} (2017) 1--37]posed two open questions whether the integral \[{\hatJ}_{k}(\alpha)=\int_0^\infty\frac{xe^{-\alpha x^2}}{e^{2\pix}-1}\,{}_1F_1(-k,3/2;2\alpha x^2)\,dx\] for $\alpha>0$ could be evaluated inclosed 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 hypergeometricfunction and a ratio of two gamma functions, together with a remainderexpressed as an integral. An upper bound on the remainder term is obtained,which is shown to be exponentially small as $k$ becomes large when $a=O(1)$.

A note on an integral of Dixit, Roy and Zaharescu