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International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 23, Pages 3849-3866
http://dx.doi.org/10.1155/IJMMS.2005.3849

A series transformation formula and related polynomials

Department of Mathematics, Ohio Northern University, Ada 45810, Ohio, USA

Received 13 October 2004; Revised 18 April 2005

Copyright © 2005 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a formula that turns power series into series of functions. This formula serves two purposes: first, it helps to evaluate some power series in a closed form; second, it transforms certain power series into asymptotic series. For example, we find the asymptotic expansions for λ>0 of the incomplete gamma function γ(λ,x) and of the Lerch transcendent Φ(x,s,λ). In one particular case, our formula reduces to a series transformation formula which appears in the works of Ramanujan and is related to the exponential (or Bell) polynomials. Another particular case, based on the geometric series, gives rise to a new class of polynomials called geometric polynomials.