Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi

Abstract

In this article, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the light of properties of partial Bell polynomials, the author establishes Taylor’s series expansions of real powers of two functions containing squares of inverse (hyperbolic) cosine functions in terms of the Stirling numbers of the first kind, presents a closed-form formula of specific partial Bell polynomials at a sequence of derivatives of a function containing the square of inverse cosine function, derives several combinatorial identities involving the Stirling numbers of the first kind, demonstrates several series representations of the circular constant Pi and its real powers, recovers Maclaurin’s series expansions of positive integer powers of inverse (hyperbolic) sine functions in terms of the Stirling numbers of the first kind, and also deduces other useful, meaningful, and significant conclusions and an application to the Riemann zeta function.

Keywords

• taylor’s series expansion
• maclaurin’s series expansion
• real power
• inverse cosine function
• inverse hyperbolic cosine function
• inverse sine function
• inverse hyperbolic sine function
• stirling number of the first kind
• combinatorial identity
• composite
• series representation
• circular constant
• partial bell polynomial
• closed-form formula
• riemann zeta function
• primary 41a58
• secondary 05a19
• 11b73
• 11b83
• 11c08
• 11s05
• 12d05
• 26a24
• 26c05
• 33b10