Appell Polynomials and Their Zero Attractors

TL;DR

This paper investigates the asymptotic behavior and zero distribution of Appell polynomials generated by entire functions with zeros, extending previous work on Euler polynomials.

Contribution

It provides a detailed analysis of the zero attractors and asymptotics of Appell polynomials associated with entire functions, including those with zeros.

Findings

Describes asymptotics of scaled Appell polynomials using zeros of the generating function

Determines the limiting distribution and density of zeros of Appell polynomials

Extends previous results on Euler polynomials to a broader class of Appell polynomials

Abstract

A polynomial family $\{p_n(x)\}$ is Appell if it is given by $\frac{e^{xt}}{g(t)} = \sum_{n=0}^\infty p_n(x)t^n$ or, equivalently, $p_n'(x) = p_{n-1}(x)$. If $g(t)$ is an entire function, $g(0)\neq 0$, with at least one zero, the asymptotics of linearly scaled polynomials $\{p_n(nx)\}$ are described by means of finitely zeros of $g$, including those of minimal modulus. As a consequence, we determine the limiting behavior of their zeros as well as their density. The techniques and results extend our earlier work on Euler polynomials.