Expansions of the exponential and the logarithm of power series and applications

Feng Qi,Xiao-Ting Shi,Fang-Fang Liu

doi:10.1007/s40065-017-0166-4

Abstract

In the paper, the authors establish explicit formulas for asymptotic and power series expansions of the exponential and the logarithm of asymptotic and power series expansions. The explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion are applied to find explicit formulas for the Bell numbers and logarithmic polynomials in combinatorics and number theory.

Highlights

Throughout this paper, we understand an empty sum to be 0 and regard an empty product as 1
We observe that the constant term did not appear in (1.2) in Theorem 1.5 and that the formulas (1.3) and (1.4) are recurrences
We will apply Theorems 4.1 and 4.2, respectively, to the Bell numbers and logarithmic polynomials which are extensively studied in combinatorics and number theory

Summary

Introduction

Throughout this paper, we understand an empty sum to be 0 and regard an empty product as 1. Let us recall definitions for an asymptotic expansion. In which the sum of the first n + 1 terms is Sn(z), is said to be an asymptotic expansion of a function f (z) for a given range of values of arg z, if the expression. The composition B(x) = eA(x) has the asymptotic expansion where b0 = 1 and. ∞ n=0 cn xn be a given asymptotic expansion. We observe that the constant term did not appear in (1.2) in Theorem 1.5 and that the formulas (1.3) and (1.4) are recurrences. 2 of this paper, we will add a constant term into (1.2) and acquire a modified version of Theorem 1.5. The function E(x) = eD(x) has the asymptotic expansion.

Explicit formulas for an and en

Expansions of the exponential and logarithm of power series

Remarks