Complete asymptotic expansion for a Durrmeyer variant of operators based on Hermite polynomials

What the math gives to ML

The paper's operator is a normalized positive kernel whose weights form an explicit compound-Poisson distribution: the sampled index is the sum of an ordinary Poisson variable and an independent even-jump Poisson variable. This gives a nonnegative, analytically controlled stochastic smoothing operator with tunable mean and variance, rather than an arbitrary noise injection. A practical transfer is to insert this kernel before selected nonnegative neural features, using the induced count-valued perturbation as a structured regularizer or uncertainty mechanism. The main opportunity is not the asymptotic expansion itself, whose coefficients are not present in the extraction, but the exact probability law and its moment control.