Raising Series to A Power

Abstract

Objective. Let be a formal power series and let . In this note, we consider the function . We find that if has a series expansion at , then its coefficients are polynomials in . The coefficients of these polynomials were found to be a weighted composition sum. Methods. The method to arrive at this representation involves logarithmic derivative and exponential representation. Findings. As a consequence of this, new identities involving partition functions and binomial coefficients were obtained. Further, a particular class of Dirichlet series is found to have the form of an exponential function. Consequently, identities involving Riemann zeta function values were obtained. Novelty. The present work generalizes a class of functions considered by D’Arcais. Divisor-sum identities involving partition functions and exponential representation of Dirichlet series of this article were new to the literature. Keywords: Polynomials, Partitions, Dirichlet Series, Divisor­Sum, Power Series