International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 23, Pages 3849-3866
doi: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 Khristo N. Boyadzhiev. 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.