Appell Polynomials and Their Zero Attractors

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, $p_n^{\prime }(x)=p_{n-1}(x)$. If $g(t)$ is an entire function, $g(0)\ne 0$, 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.

23 pages, 9 Figures