Brenke polynomials with real zeros and the Riemann Hypothesis

TL;DR

This paper characterizes when Brenke polynomials have only real zeros and explores their connection to the Laguerre-Pólya class, providing new equivalences related to the Riemann Hypothesis through real-rootedness of certain Brenke polynomial sequences.

Contribution

It extends the characterization of real zeros from Appell polynomials to Brenke polynomials and links these results to the Riemann Hypothesis.

Findings

Necessary and sufficient conditions for Brenke polynomials to have real zeros.

Identification of when Brenke polynomials belong to the Laguerre-Pólya class.

New equivalencies for the Riemann Hypothesis based on real-rooted Brenke polynomials.

Abstract

If $A(z)=\sum_{n=0}^\infty a_nz^n$ and $B(z)=\sum_{n=0}^\infty b_nz^n$ are two formal power series, with $a_n,b_n\in \mathbb{R}$, the polynomials $(p_n)_n$ defined by the generating function $$ A(z)B(xz)=\sum_{n=0}^\infty p_n(x)z^n $$ are called the Brenke polynomials generated by $A$ and associated to $B$. We say that $A\in \mathcal{R}_B$ 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 $\mathcal{R}_B=\mathcal{L}\text{-}\mathcal{P}$, where $\mathcal{L}\text{-}\mathcal{P}$ denotes the Laguerre-P\'olya class (of entire functions). These results can be considered an extension to Brenke polynomials of the Jensen, and P\'olya and Schur characterization $\mathcal{R}_{e^z}=\mathcal{L}\text{-}\mathcal{P}$, 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.