Brenke polynomials with real zeros and the Riemann Hypothesis

Abstract

If $A(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ and $B(z)={\sum }_{n=0}^{\infty }{b}_n{z}^n$ 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-Pólya class (of entire functions). These results can be considered an extension to Brenke polynomials of the Jensen, and Pólya 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.