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

TL;DR

This paper develops a general integral method to derive new power series involving central binomial coefficients, providing novel series expansions for pi and its powers, and exploring their connections with hypergeometric series.

Contribution

It introduces a new integral transformation technique applied to powers of arcsine, generating infinite classes of series involving central binomial coefficients, and links these to hypergeometric series.

Findings

New power series for pi and its powers derived

Infinite classes of series involving central binomial coefficients established

Connections with hypergeometric series discussed

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 $\pi$ and some of its positive integer powers. We also discuss several limit expressions and connections with hypergeometric series.