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

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.

arXiv comment28 pages

References in corpus (7)

• New results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams
• Derivatives of tangent function and tangent numbers
• Explicit expressions for a family of Bell polynomials and derivatives of some functions
• LSJK - a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine functions
• Several combinatorial identities derived from series expansions of powers of arcsine
• Diagonal recurrence relations for the Stirling numbers of the first kind
• A Series Expansion for Integral Powers of Arctangent

Cited by in corpus (3)

• Taylor's series expansions for real powers of functions containing squares of inverse (hyperbolic) cosine functions, explicit formulas for special partial Bell polynomials, and series representations for powers of circular constant
• Several series expansions for real powers and several formulas for partial Bell polynomials of sinc and sinhc functions in terms of central factorial and Stirling numbers of second kind
• Closed-form formulas, determinantal expressions, recursive relations, power series, and special values of several functions used in Clark--Ismail's two conjectures