Difference equations related to number theory
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 \(g(n)=1\): \begin{align*} P_n^1 (x) &= (-1)^n \binom{-x}{n} , \\ Q_n^1 (x) &= (x+1)^{n-1}x , \end{align*} finding a connection between them via a theorem of \textit{H. Poincaré} [Am. J. Math. 7, 203--258 (1885; JFM 17.0290.01)]; they eventually suggest a further investigation under arbitrary conditions. For the entire collection see [Zbl 1467.39001].
scientific article; zbMATH DE number 7388621