An asymptotic formula for the exponential polynomials and a central limit theorem for their coefficients

On Asymptotics for the Signless Noncentral q‐Stirling Numbers of the First Kind

“…Having obtained (21) assuming that ψ is an independent variable we set, again, ψ = [g (r )] −1/2 . However, this will only be legitimate if we show that this point lies within the domain of convergence of (21) (see [6]). In other words we must prove that |b k (r, φ) ψ k k !…”

“…From (19) and (21), we have along with the fact that → ∞ as n → ∞ justify the replacement of by ∞ (see [6]). Thus…”

“…In this section, we present the details of the asymptotic analysis of the signless noncentral q-Stirling numbers of the first kind using an approach of the saddle point method inspired from [7], [6], and [9].…”

“…Since P(|φ|) is a polynomial in |φ| the integral in (46) exists and |R N | = O(n −(N +1)/2 ). Noting that b 2k +1 (r, φ), as polynomials in φ, contain only odd powers (see [6]), Equation (44) reduces to…”

“…(For complete surveys on this technique, the reader is referred to the works of Odlyzko [3] and Flajolet and Sedgewick [4]. For interesting applications see, for example, the papers of Moser and Wyman [5], Charalambides and Kyriakoussis [6], and Kirousis et al [7]. In the last paper the authors considered the q-Binomial coefficients having closely limiting behavior with the signless noncentral q-Stirling numbers of the first kind).…”