Algebraic independence of the generating functions of the stern polynomials and their twisted analogues
type: Item
instance of: scholarly article
title: Algebraic independence of the generating functions of the stern polynomials and their twisted analogues
zbMATH Open document ID: 1323.11053
author: Peter Bundschuh
author: Keijo Väänänen
published in: Archiv der Mathematik
publication date: 2015-09-01T00:00:00Z
review text: Let \(\{ a_n(x)\}_{n=0}^\infty\) and \(\{ b_n(x)\}_{n=0}^\infty\) be two recurrent sequences of polynomials such that \(a_0(x)=b_0(x)=0\), \(a_1(x)=b_1(x)=1\) and \(a_{2n}(x)=a_n(x^2)\), \(b_{2n}(x)=-b_n(x^2)\), \(a_{2n+1}(x)=xa_n(x^2)+ a_{n+1}(x^2)\), \(b_{2n+1}(x)=-(xb_n(x^2)+ b_{n+1}(x^2))\) for all \(n\in\mathbb Z^+\). Set \[ A(x,z)=\sum_{n=0}^\infty a_{n+1}(x)z^n, \] \[ B(x,z)=\sum_{n=0}^\infty b_{n+1}(x)z^n\, \text{ and}\, P(x,z)=1+z+xz^2. \] Let \(\mathbb D\) is an open unit disk. Then for \(\alpha,\beta\in \overline {\mathbb Q}^\times\cap \mathbb D\) we have that \(P(\alpha^{2^k},\beta^{2^k})\not= 0\) holds for any \(k\in\mathbb N_0\) if and only if the numbers \(A(\alpha,\beta)\) and \(B(\alpha,\beta)\) are algebraically independent over \(\mathbb Q\). The proof is in the spirit of Mahler.
reviewed by: Jaroslav Hančl
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11J91
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 39B32
zbMATH DE Number: 6477365
zbMATH Keywords: Stern sequence and twisted version
zbMATH Keywords: polynomial analogues
zbMATH Keywords: generating functions
zbMATH Keywords: transcendence and algebraic independence
zbMATH Keywords: Mahler's method
MaRDI profile type: Publication
full work available at URL: s00013-015-0789-7
OpenAlex ID: W965210270
cites work: TRANSCENDENCE AND ALGEBRAIC INDEPENDENCE OF SERIES RELATED TO STERN'S SEQUENCE
cites work: Algebraic independence of the generating functions of Stern's sequence and of its twist
cites work: THE TRANSCENDENCE OF SERIES RELATED TO STERN'S DIATOMIC SEQUENCE
cites work: A POLYNOMIAL ANALOGUE TO THE STERN SEQUENCE
cites work: On the algebraic independence of holomorphic solutions of certain functional equations and their values
cites work: Mahler functions and transcendence
DOI: 10.1007/S00013-015-0789-7
description: scientific article; zbMATH DE number 6477365