Zeros of Stern polynomials in the complex plane
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918188080431104 |
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| author | Altizio, David |
| author_facet | Altizio, David |
| contents | The classical Stern sequence of positive integers was extended to a polynomial sequence $S_n(λ)$ by Klavžar et. al. by defining $S_0(λ) = 0$, $S_1(λ) = 1$, and $$S_{2n}(λ) = λS_n(λ),\quad S_{2n+1}(λ) = S_n(λ) + S_{n+1}(λ).$$ Dilcher et. al. conjectured that all roots of $S_n(λ)$ lie in the half-plane $\{\operatorname{Re} w < 1\}$. We make partial progress on this conjecture by proving that $\{|w-2| \leq 1\}\subseteq\mathbb C$ does not contain any roots of $S_n(λ)$. Our proof uses the Parabola Theorem for convergence of complex continued fractions. As a corollary, we establish a conjecture of Ulas and Ulas by showing that $S_p(λ)$ is irreducible in $\mathbb Z[λ]$ whenever $p$ is a positive prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03847 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeros of Stern polynomials in the complex plane Altizio, David Number Theory Combinatorics 26C10, 30B70 (Primary) 11R09 (Secondary) The classical Stern sequence of positive integers was extended to a polynomial sequence $S_n(λ)$ by Klavžar et. al. by defining $S_0(λ) = 0$, $S_1(λ) = 1$, and $$S_{2n}(λ) = λS_n(λ),\quad S_{2n+1}(λ) = S_n(λ) + S_{n+1}(λ).$$ Dilcher et. al. conjectured that all roots of $S_n(λ)$ lie in the half-plane $\{\operatorname{Re} w < 1\}$. We make partial progress on this conjecture by proving that $\{|w-2| \leq 1\}\subseteq\mathbb C$ does not contain any roots of $S_n(λ)$. Our proof uses the Parabola Theorem for convergence of complex continued fractions. As a corollary, we establish a conjecture of Ulas and Ulas by showing that $S_p(λ)$ is irreducible in $\mathbb Z[λ]$ whenever $p$ is a positive prime. |
| title | Zeros of Stern polynomials in the complex plane |
| topic | Number Theory Combinatorics 26C10, 30B70 (Primary) 11R09 (Secondary) |
| url | https://arxiv.org/abs/2511.03847 |