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arXiv · 1705.04723

Generalized Log-sine integrals and Bell polynomials

Derek Orr
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Abstract

In this paper, we investigate the integral of $x^n\log^m(\sin(x))$ for natural numbers $m$ and $n$. In doing so, we recover some well-known results and remark on some relations to the log-sine integral $\operatorname{Ls}_{n+m+1}^{(n)}(\theta)$. Later, we use properties of Bell polynomials to find a closed expression for the derivative of the central binomial and shifted central binomial coefficients in terms of polygamma functions and harmonic numbers.

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Derek Orr. 2017-05-12. Generalized Log-sine integrals and Bell polynomials. https://doi.org/10.1016/j.cam.2018.08.026

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