The Ramanujan Journal

, Volume 46, Issue 1, pp 161–171 | Cite as

A family of polylog-trigonometric integrals

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Abstract

In this paper, we study the sequences
$$\begin{aligned} I_n=\int _0^1\mathrm {Li}_n(\sin \pi x)\mathrm {d}x\quad \text{ and }\quad J_n=\int _0^1\mathrm {Li}_n(\cos \pi x)\mathrm {d}x, \end{aligned}$$
where \(\mathrm {Li}_n\) is the nth polylogarithm function. Among others, we determine their generating functions, asymptotic behaviour and their connection to the well-known log-sine integrals
$$\begin{aligned} S_n=(-1)^n\int _0^1\log ^n(\sin \pi x)\mathrm {d}x. \end{aligned}$$
With the help of the explicit forms of \(I_n\) and \(J_n\), we deduce closed-form evaluations for a number of polylog-trigonometric definite integrals.

Keywords

Log-sine integrals Polylogarithm functions Gamma function Hypergeometric function 

Mathematics Subject Classification

33B15 

Notes

Acknowledgements

The author is grateful to the unknown referee for the detailed review of the manuscript.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsNanjing University of Information Science and TechnologyNanjingPeople’s Republic of China

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