A note on the reducibility of special infinite series

Abstract

In this article we present how to reduce the evaluation of the special infinite series $\text{S}{}_{\mathrm{n}}\text{=∑}{}^{\mathrm{\infty }}{}_{\mathrm{k=1}}\text{k}{}^{\mathrm{n}}\text{(k+1)!}$, n=1,2,3,… which are very time consuming in Computer Algebra Systems (CAS). An algorithm is given for this purpose. Based on the new algorithm, a fast and reliable MAPLE procedure is designed and S₁,S₂,…,S₅₀ are given as sample output of this procedure together with the cpu time.

Introduction

The Stirling numbers of the second kind [1], [2] describe the number of ways a set with n elements can be partitioned into k disjoint, non-empty subsets. These numbers usually denoted by S(n,k). In [3] it is shown that by introducing the operator

$$\text{θ≡xD≡x}\text{d}\text{d}\text{x}$$

then we have

$$\text{θ}{}^{\mathrm{n}}\text{e}{}^{\mathrm{x}}\text{=∑}{}_{\mathrm{k=1}}{}^{\mathrm{\infty }}\text{k}{}^{\mathrm{n}}\text{x}{}^{\mathrm{k}}\text{k!}$$

and

$$\text{θ}{}^{\mathrm{n}}\text{e}{}^{\mathrm{x}}\text{=}\text{e}{}^{\mathrm{x}}\text{∑}{}_{\mathrm{k=1}}{}^{\mathrm{n}}\text{S(n,k)x}{}^{\mathrm{k}}\text{.}$$

The motivation of the current paper is to use , in order to develop an algorithm to evaluate the infinite series of the form $\text{S}{}_{\mathrm{n}}\text{=∑}{}^{\mathrm{\infty }}{}_{\mathrm{k=1}}\text{k}{}^{\mathrm{n}}\text{(k+1)!}$, n=1,2,3,… efficiently (for n=0 it is known that S₀=e−2). The main results are presented in Section 2.