A series transformation formula and related polynomials

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

Polynomials related to harmonic numbers and evaluation of harmonic number series II

“…In section 5 we deal with the general geometric polynomials. Exponential and geometric polynomials are connected by the following integral relation [6] …”

“…The particular case r = 0 in the Theorem 3 refers to the Theorem 4.1 of Boyadzhiev [6]. Therefore from now on we consider the case r ≥ 1.…”

The generalized Touchard polynomials revisited

“…Let us point out that in [1] certain analogs of the Touchard (or exponential) polynomials are considered.…”

“…The Touchard polynomials (also called exponential polynomials) may be defined (see, e.g., [1,2,6,16,17]) for n 2 N by T n ðxÞ :¼ e Àx x d dx n e x : ð1:1Þ…”

Lah numbers, Laguerre polynomials of order negative one, and the nth derivative of exp(1/x)

“…This will reveal interesting connections of Lah numbers to Laguerre polynomials and also to Stirling numbers. In Section 4 we present a new formula for D n e c x p in terms of the exponential polynomials ϕ n (x) considered in [4] and [5].…”