Powers of the arcsine and infinite classes of series involving central binomial coefficients

TL;DR

The paper develops an integral-transform framework to generate infinite families of central-binomial- coefficient series from $\arcsin x$ and its powers up to the fourth power. A general moment identity for $f(x)$ and a differential operator $\mathcal{D}$ underpin a constructive approach that yields explicit closed forms for sums involving $\binom{2k}{k}$ and powers of $x$, including $x=1$ limits that converge to $\pi$, $\pi^2$, $\pi^3$, and $\pi^4$. It provides detailed results for each power $\arcsin x$, delivering even/odd $n$ formulas, limit expressions, and special $x$ values, along with connections to hypergeometric functions. The work thus furnishes new rapidly convergent series representations for fundamental constants and reveals rich links between binomial-series, moments of $\arcsin$, and hypergeometric identities with potential implications for number theory and computational mathematics.

Abstract

A general integral expression to transform power series is applied to $\arcsin{x}$ and its positive integer powers. We concentrate on the first to the fourth powers and obtain infinite classes of new power series involving central binomial coefficients. Specializing the variable to appropriate simple values leads to different classes of series expansions for $π$ and some of its positive integer powers. We also discuss several limit expressions and connections with hypergeometric series.

Theorems & Definitions (43)

• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3
• proof : Proof of Corollary \ref{['cor:3.3']}