AI 中文总结
本文扩展了移位渐近展开的Appell多项式构造，建立了Borel-Laplace表示，推导了Mills-Hermite展开的余项估计与有限差分公式，将其应用于非中心高斯尾展开，并得到Stieltjes矩序列对应的双侧界层级。
AI 中文摘要
Burić、Elezović和Vukšić的定理指出，将渐近展开$f(x)\backsim\textstyle\bigcup(-1)^na_nx^{-n-1}$中的自变量平移为$f(x+t)$，会将常系数$a_n$替换为由$(a_n)$生成的Appell多项式$R_n(t)$。其动机示例来自伽马函数和多伽马函数，其中出现了Bernoulli多项式。我们将该构造扩展到该情形之外，并确定Borel-Laplace表示$L_A(x)=\textstyle\bigcup_0^\nu e^{-xs}A(-s)\text{d}s$，只要积分存在。对于高斯核$A(-s)=e^{-s^2/2}$，该表示给出了Mills-Hermite展开$M(x+t)\backsim\bigcup(-1)^n\text{He}_n(t)x^{-n-1}$，扩展了经典标量展开。我们建立了显式余项估计，并推导了仍在Hermite多项式代数中的有限差分公式。作为应用，我们得到了非中心高斯尾的大阈值展开，其多项式依赖于均值。若$A(-s)$本身是正测度的拉普拉斯变换，则系数构成Stieltjes矩序列。相关的Padé收敛给出了系统的双侧界层级，其中连续的下、上近似依次纳入矩。
英文摘要
A theorem of Burić, Elezović and Vuk\v sić 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 Padé convergents give a systematic hierarchy of two-sided bounds in which successive lower and upper approximants incorporate the moments one at a time.
