Appell Polynomials in Shifted Asymptotic Expansions: the Mills ratio, Hermite polynomials, and Stieltjes bounds

Abstract

A theorem of Buri\'c, Elezovi\'c and Vuk\v si\'c states that translating the argument in an asymptotic expansion, $f(x)\sim\sum(-1)^na_nx^{-n-1}$, to $f(x+t)$, replaces the constant coefficients $a_n$ by the Appell polynomials $R_n(t)$ generated by $(a_n)$. Their motivating examples came from the gamma and polygamma functions, where Bernoulli polynomials occur. We extend this construction beyond that setting and identify the Borel--Laplace representation $L_A(x)=\int_0^\infty e^{-xs}A(-s)\,\dd s$, whenever the integral exists. For the Gaussian kernel $A(-s)=e^{-s^2/2}$, this representation gives the \emph{Mills--Hermite} expansion $M(x+t)\sim\sum(-1)^n\He_n(t)x^{-n-1}$, extending the classical scalar expansion. We establish an explicit remainder estimate and derive finite-difference formulas that remain in the Hermite polynomial algebra. As an application, we obtain a large-threshold expansion of the non-central Gaussian tail with explicit polynomial dependence on the mean. If $A(-s)$ is itself the Laplace transform of a positive measure, then the coefficients form a Stieltjes moment sequence. The associated Pad\'e convergents give a systematic hierarchy of two-sided bounds in which successive lower and upper approximants incorporate the moments one at a time.