Brenke polynomials with real zeros and the Riemann Hypothesis

2024/05/29 by Antonio J. Durán, Durán, Antonio J. · 2 citations
Mathematics · #11M26 #26C10 #30C15 #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical functions and polynomials #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2405.18940

openalex publication_date 2024/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

If A(z)=∑_n=0^∞ a_nzⁿ and B(z)=∑_n=0^∞ b_nzⁿ are two formal power series, with a_n,b_n∈ ℝ, the polynomials (p_n)_n defined by the generating function A(z)B(xz)=∑_n=0^∞ p_n(x)zⁿ are called the Brenke polynomials generated by A and associated to B. We say that A∈ R_B if the Brenke polynomials (p_n)_n have only real zeros. Among other results, in this paper we find necessary and sufficient conditions on B such that R_B=L-P, where L-P denotes the Laguerre-Pólya class (of entire functions). These results can be considered an extension to Brenke polynomials of the Jensen, and Pólya and Schur characterization R_e^z=L-P, for Appell polynomials. When applying our results to a relative of the Riemann zeta function, we find new equivalencies for the Riemann Hypothesis in terms of real-rootedness of some sequences of Brenke polynomials.

Cited by

• Zeros of linear combinations of Laguerre polynomials
• Zeros of linear combinations of Hermite polynomials

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