Difference equations related to number theory
type: Item
instance of: scholarly article
title: Difference equations related to number theory
zbMATH Open document ID: 1472.39032
DOI: 10.1007/978-3-030-60107-2_11
author: Bernhard Heim
author: Markus Neuhauser
publication date: 2021-08-31T00:00:00Z
review text: As part of a collection on difference equations and applications [Zbl 1467.39001], this article requires a high level of knowledge on the subject. The authors recall the Dedekind's \(\eta\)-function they already studied in [Res. Math. Sci. 7, No. 1, Paper No. 3, 8 p. (2020; Zbl 1472.11122)], clarifying how its powers are linked to a polynomial defined recursively as follows: \begin{align*} P_1(x) &= x , \\ P_n(x) &= \frac{x}{n} \left( \sigma(n) + \sum_{k=1}^{n-1} \sigma(k) P_{n-k}(x) \right) , \end{align*} where \(x \in \mathbb{C}\) and \(\sigma(k)\) is the sum of the divisors of \(k\). After remarking the importance of \(P_n(x)\) in number theory, the authors acknowledge its irreducibility to a recurrence relation of bounded length and they propose to generalize it through the arithmetic functions \(g,h\): \begin{align*} P_1^{g,h}(x) &= x , \\ P_n^{g,h}(x) &= \frac{x}{h(n)} \left( g(n) + \sum_{k=1}^{n-1} g(k) P_{n-k}^{g,h}(x) \right) , \end{align*} with \(g: \mathbb{N} \rightarrow \mathbb{C}\), \(h: \mathbb{N} \rightarrow \mathbb{R}\), and \(g(1)=h(1)=1\). Beside a quick mention to the classical orthogonal polynomials as solutions of a specific differential equation, the authors distinguish the following subcases: \begin{align*} P_n^g (x) &= \frac{x}{n} \left( g(n) + \sum_{k=1}^{n-1} g(k) P_{n-k}^{g}(x) \right) , \\ Q_n^g (x) &= x \left( g(n) + \sum_{k=1}^{n-1} g(k) Q_{n-k}^{g}(x) \right) , \end{align*} related, respectively, to the associated Laguerre polynomials and to the Chebyshev polynomials of the second kind. The authors employ some of their previous results, obtained in collaboration with \textit{R. Tröger} [J. Difference Equ. Appl. 26, No. 4, 510--531 (2020; Zbl 1456.30010)], in order to analyze the limiting behavior of the main sequence \(\left( P_n^{g,h}(x) \right)_{n \in \mathbb{N}}\) and of the subsequence \(\left( Q_n^g (x) \right)_{n \in \mathbb{N}}\). Then the authors focus on the recurrence relations of \(P_n^g (x)\) and \(Q_n^g (x)\) for \(g(n)=1\…
Mathematics Subject Classification ID: 39A60
Mathematics Subject Classification ID: 39-06
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 11Z05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 7388621
zbMATH Keywords: difference equations
zbMATH Keywords: arithmetic functions
zbMATH Keywords: classical orthogonal polynomials
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: Dedekind's \(\eta\)-function
zbMATH Keywords: polynomials
zbMATH Keywords: recurrences
MaRDI profile type: Publication
full work available at URL: 978-3-030-60107-2 11
OpenAlex ID: W3118477243
cites work: Q4218535
cites work: The Classical Orthogonal Polynomials
cites work: Q4679023
cites work: Recurrence relations for polynomials obtained by arithmetic functions
cites work: The Dedekind eta function and D'Arcais-type polynomials
cites work: Zeros of recursively defined polynomials
cites work: The vanishing of Ramanujan's function \(\tau(n)\)
cites work: Q4449110
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description: scientific article; zbMATH DE number 7388621