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

What the math gives to ML

The paper provides a systematic way to turn shifted Gaussian-tail asymptotics into Hermite-polynomial expansions, rather than recomputing a separate scalar expansion for every mean shift. The transferable asset is a compact, recursively computable approximation whose coefficients expose explicit dependence on the shift and whose truncation behavior is controlled in the large-threshold regime. This can be used as a specialized Gaussian-tail or probit primitive in neural losses or diffusion guidance when many evaluations occur at large standardized thresholds. The opportunity is narrow but concrete: benchmark the Hermite approximation against exact special-function implementations and rational tail approximations, with fallback to the exact primitive outside its asymptotic regime.