Generalized Log-sine integrals and Bell polynomials

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 ${\mathrm{Ls}}_{n+m+1}^{(n)}(\theta )$. 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.

19 pages

References in corpus (1)

• Generalized rational zeta series for $\zeta (2n)$ and $\zeta (2n+1)$

Cited by in corpus (1)

• Alternating multiple zeta values, and explicit formulas of some Euler-Apery-type series