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

2025/12/04 by Karl Dilcher, Dilcher, Karl, Christophe Vignat +1
Mathematics · #Advanced Mathematical Identities #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical functions and polynomials #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2512.05260

openalex publication_date 2025/12/04 · openalex created_date 2025/12/09 · openalex updated_date 2026/07/28

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.

Citations

• Powers of the arcsine and infinite classes of series involving central binomial coefficients
• Further Classes of Series Involving Central Binomial Coefficients
• Generating functions of multiple t-star values
• New properties of multiple harmonic sums modulo p and p-analogues of Leshchiner's series

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• Sums of arctangents and sums of products of arctangents
• From Basel Problem to multiple zeta values
• Infinite arctangent sums involving Fibonacci and Lucas numbers
• Infinite product of power series