  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.
