Brenke polynomials with real zeros and the Riemann Hypothesis
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916265070690304 |
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| author | Durán, Antonio J. |
| author_facet | Durán, Antonio J. |
| contents | 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-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_18940 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Brenke polynomials with real zeros and the Riemann Hypothesis Durán, Antonio J. Classical Analysis and ODEs Complex Variables Number Theory 30C15, 11M26, 26C10 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-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. |
| title | Brenke polynomials with real zeros and the Riemann Hypothesis |
| topic | Classical Analysis and ODEs Complex Variables Number Theory 30C15, 11M26, 26C10 |
| url | https://arxiv.org/abs/2405.18940 |