A novel advancement in the study of Appell polynomials via Pad\`e rational approximants

Abstract

The use of approximants of Pad\`e type are employed to develop a method aimed at opening new perspectives in the theory of Appell polynomials $a_n(x)$, specified by the generating function \sum_{n=0}^{\infty} \frac{t^n}{n!} a_n(x) = A(t) e^{xt}. In this article, the expansion of amplitude $A(t)$ of the Appell polynomials family in terms of rational approximants yields the possibility of determining the approximation of the $a_n(x)$ in terms of other special polynomials. Application of this approach to Hermite polynomials yields highly accurate approximations in terms of truncated exponential polynomials. Further, monomiality conditions are explored and formalism is extended to consider the Pad\'e approximants within the context of umbral notation.