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

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 $\pi$.