Appell Polynomials and Their Zero Attractors

2008/09/08 by Robert P. Boyer William M. Y. Goh, Goh, Robert P. Boyer William M. Y.
Mathematics · #05C38 #15A15 #Advanced Mathematical Identities #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics #Functional Equations Stability Results #Meromorphic and Entire Functions #math.CO #math.CV #msc:05C38 #msc:15A15

paper · pdf · doi:10.48550/arxiv.0809.1266

23 pages, 9 Figures

arxiv created 2008/09/08 · openalex publication_date 2008/09/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A polynomial family \p_n(x)\ is Appell if it is given by \frace^xtg(t) = ∑_n=0^∞ p_n(x)tⁿ or, equivalently, p_n'(x) = p_n-1(x). If g(t) is an entire function, g(0)≠ 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.

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