Power Series with Binomial Sums and Asymptotic Expansions

TL;DR

This paper investigates power series with binomial coefficients, providing methods for series summation, transformation, and asymptotic expansions involving special polynomials and Stirling numbers.

Contribution

It introduces new techniques for asymptotic expansions and series transformations using binomial sums and special polynomials.

Findings

Derived asymptotic expansions involving Bernoulli and Euler polynomials

Established connections to Euler series transformations

Developed methods for series summation and transformation

Abstract

This paper is a study of power series, where the coefficients are binomial expressions (iterated finite differences). Our results can be used for series summation, for series transformation, or for asymptotic expansions involving Stirling numbers of the second kind. In certain cases we obtain asymptotic expansions involving Bernoulli polynomials, poly-Bernoulli polynomials, or Euler polynomials. We also discuss connections to Euler series transformations and other series transformation formulas.