On series expansions of zeros of the deformed exponential function
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916505891897344 |
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| author | Kuznetsov, Alexey |
| author_facet | Kuznetsov, Alexey |
| contents | For $q \in (0, 1)$, the deformed exponential function $f(x) = \sum_{n \geq 1} x^n q^{n(n-1)/2}/n!$ is known to have infinitely many simple and negative zeros $\{x_k(q)\}_{k \geq 1}$. In this paper, we analyze the series expansions of $-x_k(q)/k$ and $k/x_k(q)$ in powers of $q$. We prove that the coefficients of these expansions are rational functions of the form $P_n(k)/Q_n(k)$ and $\widehat{P}_n(k)/Q_n(k)$, where $Q_n(k) \in {\mathbb Z}[k]$ is explicitly defined and the polynomials $P_n(k), \widehat{P}_n(k)\in {\mathbb Z}[k]$ can be computed recursively. We provide explicit formulas for the leading coefficients of $P_n(k)$ and $\widehat{P}_n(k)$ and compute the coefficients of these polynomials for $n \leq 300$. Numerical verification shows that $P_n(k)$ and $\widehat{P}_n(k)$ take non-negative values for all $k \in \mathbb{N}$ and $n\le 300$, offering further evidence in support of conjectures by Alan Sokal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02462 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On series expansions of zeros of the deformed exponential function Kuznetsov, Alexey Classical Analysis and ODEs Combinatorics Complex Variables 30C15, 30E15, 34K06 For $q \in (0, 1)$, the deformed exponential function $f(x) = \sum_{n \geq 1} x^n q^{n(n-1)/2}/n!$ is known to have infinitely many simple and negative zeros $\{x_k(q)\}_{k \geq 1}$. In this paper, we analyze the series expansions of $-x_k(q)/k$ and $k/x_k(q)$ in powers of $q$. We prove that the coefficients of these expansions are rational functions of the form $P_n(k)/Q_n(k)$ and $\widehat{P}_n(k)/Q_n(k)$, where $Q_n(k) \in {\mathbb Z}[k]$ is explicitly defined and the polynomials $P_n(k), \widehat{P}_n(k)\in {\mathbb Z}[k]$ can be computed recursively. We provide explicit formulas for the leading coefficients of $P_n(k)$ and $\widehat{P}_n(k)$ and compute the coefficients of these polynomials for $n \leq 300$. Numerical verification shows that $P_n(k)$ and $\widehat{P}_n(k)$ take non-negative values for all $k \in \mathbb{N}$ and $n\le 300$, offering further evidence in support of conjectures by Alan Sokal. |
| title | On series expansions of zeros of the deformed exponential function |
| topic | Classical Analysis and ODEs Combinatorics Complex Variables 30C15, 30E15, 34K06 |
| url | https://arxiv.org/abs/2412.02462 |