Maclaurin's series expansions for positive integer powers of inverse (hyperbolic) sine and related functions, specific values of partial Bell polynomials, and two applications

TL;DR

This paper derives new Maclaurin series expansions for powers of inverse trigonometric and hyperbolic functions, applies these to find closed-form formulas for Bell polynomial values, and explores series representations of Pi and related constants.

Contribution

The paper introduces novel Maclaurin series expansions for inverse sine, hyperbolic sine, tangent, and hyperbolic tangent functions, linking them with Stirling numbers and Bell polynomials, and applies these to series representations of Pi.

Findings

Derived series expansions for inverse functions involving Stirling numbers

Established closed-form formulas for partial Bell polynomial values

Presented series representations of Pi and its powers

Abstract

In the paper, the authors establish Maclaurin's series expansions and series identities for positive integer powers of the inverse sine function, for positive integer powers of the inverse hyperbolic sine function, for the composite of incomplete gamma functions with the inverse hyperbolic sine function, for positive integer powers of the inverse tangent function, and for positive integer powers of the inverse hyperbolic tangent function, in terms of the first kind Stirling numbers and binomial coefficients, apply the newly established Maclaurin's series expansion for positive integer powers of the inverse sine function to derive a closed-form formula for specific values of partial Bell polynomials and to derive a series representation of the generalized logsine function, and deduce several combinatorial identities involving the first kind Stirling numbers. Some of these results simplify and unify some known ones. All of these newly established Maclaurin's series expansions of positive integer powers of the inverse (hyperbolic) sine and tangent functions can be used to derive infinite series representations of the circular constant Pi and of positive integer powers of Pi.