Maclaurin series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions and for incomplete gamma functions with applications

Abstract

Abstract. In the paper, the authors establish nice Maclaurin series expansions and series identities for powers of the inverse sine function, for powers of the inverse hyperbolic sine function, for composites of incomplete gamma functions with the inverse hyperbolic sine function, for powers of the inverse tangent function, and for powers of the inverse hyperbolic tangent function, in terms of the first kind Stirling numbers, binomial coefficients, and multiple sums, apply the nice Maclaurin series expansion for powers of the inverse sine function to derive an explicit formula for special values of the second kind 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 the Maclaurin series expansions of powers of the inverse trigonometric functions can be used to derive infinite series representations of corresponding powers of the constant Pi.