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

TL;DR

The paper develops a generating-function framework to evaluate moments of arbitrary powers of $\arcsin x$ via the integrals $I_q^{(n)}(x)=\int_0^x t^{n}(\arcsin t)^q\,dt$, yielding explicit closed forms for all $n\ge 0$, $q\ge 1$ and revealing four parity-based cases. By expressing these integrals through an exponential generating function and Chebyshev polynomials, the work connects arcsin moments to series involving central binomial coefficients and the multiple harmonic sums $G_p(k)$ and $H_p(k$,) producing numerous special-case identities and limit expressions. These results generate a wealth of new series evaluations and π-power limits, tying the arcsin framework to Dirichlet L-series, Bernoulli and Euler numbers, and classical constants like $\pi$. The methodology thus provides a unified route to an extensive family of $\pi$-related identities and deepens the link between arcsin moments and analytic number theory.

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 $π$. Finally, we obtain limit expressions for arbitrary powers of $π$.

Theorems & Definitions (36)

• Lemma 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof