Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi

Abstract In this article, by virtue of expansions two finite products finitely many square sums, with the aid series composite functions (hyperbolic) sine and cosine inverse functions, in light properties partial Bell polynomials, author establishes Taylor’s real powers containing squares terms Stirling numbers first kind, presents a closed-form formula specific polynomials at sequence derivatives function function, derives several combinatorial identities involving demonstrates representations circular constant Pi its powers, recovers Maclaurin’s positive integer also deduces other useful, meaningful, significant conclusions an application to Riemann zeta function.