Specific values of partial Bell polynomials and series expansions for real powers of functions and for composite functions

Abstract

Starting from Maclaurin’s series expansions for positive integer powers of analytic functions, the authors derive an explicit formula for specific values of partial Bell polynomials, present a general term of Maclaurin’s series expansions for real powers of analytic functions, obtain Maclaurin’s series expansions of some composite functions, recover Maclaurin’s series expansions for real powers of inverse sine function and sinc function, recover a combinatorial identity involving the falling factorials and the Stirling numbers of the second kind, deduce an explicit formula of the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind, recover an explicit formula of the Bernoulli numbers in terms of the Stirling numbers of the second kind, recover an explicit formula of the Bell numbers in terms of the Stirling numbers of the second kind, reformulate three specific partial Bell polynomials in terms of central factorial numbers of the second kind, and present some Maclaurin’s series expansions and identities related to the Euler numbers and their generating function.