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

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.