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In the paper, the authors establish Maclaurin's series expansions and seriesidentities for positive integer powers of the inverse sine function, forpositive integer powers of the inverse hyperbolic sine function, for thecomposite of incomplete gamma functions with the inverse hyperbolic sinefunction, for positive integer powers of the inverse tangent function, and forpositive integer powers of the inverse hyperbolic tangent function, in terms ofthe first kind Stirling numbers and binomial coefficients, apply the newlyestablished Maclaurin's series expansion for positive integer powers of theinverse sine function to derive a closed-form formula for specific values ofpartial Bell polynomials and to derive a series representation of thegeneralized logsine function, and deduce several combinatorial identitiesinvolving the first kind Stirling numbers. Some of these results simplify andunify some known ones. All of these newly established Maclaurin's seriesexpansions of positive integer powers of the inverse (hyperbolic) sine andtangent functions can be used to derive infinite series representations of thecircular constant Pi and of positive integer powers of Pi.

applicable analysis and discrete mathematics

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