Integrals involving arbitrary powers of the arcsine, with applications to infinite series

TL;DR

This paper develops methods to evaluate moments of powers of the arcsine function, leading to new results on infinite series, binomial coefficients, harmonic sums, and limits involving powers of pi.

Contribution

It introduces a systematic approach to compute moments of arbitrary powers of the arcsine and applies this to evaluate complex series and limits involving pi.

Findings

Explicit formulas for moments of powers of the arcsine

Evaluation of infinite series with binomial coefficients and harmonic sums

Limit expressions for powers of pi

Abstract

Using appropriate power series evaluations, we determine all moments of arbitrary positive powers of the arcsine. As consequences we evaluate several doubly infinite classes of power series involving central binomial coefficients and generalized multiple harmonic sums. By specializing the variable involved, we then evaluate classes of numerical sequences, mostly in terms of powers of $\pi$. Finally, we obtain limit expressions for arbitrary powers of $\pi$.