Linear relations between pattern sequences in a \(\langle q, r\rangle\)-numeration system
type: Item
instance of: scholarly article
title: Linear relations between pattern sequences in a \(\langle q, r\rangle\)-numeration system
zbMATH Open document ID: 1249.11014
published in: Acta Mathematica Hungarica
publication date: 2012-02-14T00:00:00Z
review text: The authors of this paper discuss some problems in the theory of Numeration. Every possitive integer \(n\) can be represented uniquely in the form \(n=\sum^{k-1}_{i=0}b_{i}q^{i}\), where \(q\geq 2\) and \(r\) are integers, \(0\leq r\leq q-2\); \(b_{i}\in \Sigma_{q,r}:=\left\{-r, -r+1, \ldots , 0, 1, \ldots, q-r-1\right\}\). Here \(b_{i}\) are the digits in the \(\left\langle q, r\right\rangle\)-expansion of \(n\). (\(\left\langle q, 0\right\rangle\) is the ordinary numeration system, for instance \(q=10\) etc.) Every finite string of element in \(\Sigma_{q,r}\) is called pattern (or word) over \(\Sigma_{q,r}\). The set of all finite nonempty words is denoted by \(\Sigma^{*}_{q,r}\). The length \(| w| \) is by definition the number of the elements in \(w\). \(w^{k}\) means \(ww\ldots w\) (\(k\) times); \(w^{0}\) means an empty set; \(e_{q,r}(w;n)\) means the number of occurrences of \(w\) in the \(\left\langle q, r\right\rangle\)-expansion of \(n\),\(e_{q,r}(w;0)=0\) for any \(w\in\Sigma^{*}_{q,r}\). Finally, the authors define generating function of the pattern sequence \({e_{q,r}(w;n)}_{n\geq 0}\) for \(w\in \Sigma^{*}_{q,r}\) by \[ f(w;z)=\sum_{n\geq 0}e_{q,r} (w;n)z^{n} \] The authors prove six theorems. The first is (Theorem 1): let \(m\geq2\) and \(w_{1},\ldots,w_{m}\in\Sigma^{*}_{q,r}\) with \(l=\max_{1\leq i\leq m}| w_{i}| \). Then the following five statements are equivalent: (i) \(f(w_{1};z),\ldots, f(w_{m};z)\) are algebraically dependent over \({\mathbb C}(z)\). (ii) \(f(w_{1};z),\ldots, f(w_{m};z)\) are algebraically dependent over \({\mathbb C}\;\text{ mod }\, {\mathbb C}(z)\). (iii) There exist \(c_{1},\ldots, c_{m}\in {\mathbb C}\), not all zero, such that \(\left\{\sum^{m}_{i=1}c_{i}e_{q,r}(w_{i};n)\right\}_{n\geq0}\) is a linear recurrence sequence. (iv) There exist \(c_{i},\ldots, c_{m}\in {\mathbb C}\), not all zero, such that \(\left\{\sum^{m}_{i=1}c_{i} e_{q,r}(w_{i};n)\right\}_{n\geq0}\) is a purely periodic sequence with a period \(q^{l-1}\). …
reviewed by: Tonko Tonkov
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11J85
zbMATH DE Number: 6006426
zbMATH Keywords: pattern sequence
zbMATH Keywords: block counting
zbMATH Keywords: digit
zbMATH Keywords: \(\langle q, r\rangle\)-numeration system
zbMATH Keywords: positional number system
zbMATH Keywords: algebraic independence
author: Iekata Shiokawa
author: Yohei Tachiya
MaRDI profile type: Publication
full work available at URL: s10474-011-0082-6
OpenAlex ID: W1964044875
cites work: Automatic Sequences
cites work: Subblock Occurrences in the q-Ary Representation of n
cites work: Subblock Occurrences in Positional Number Systems and Gray code Representation
cites work: Q3935355
cites work: Algebraic independence results related to \(\langle q,r\rangle\)-number systems
cites work: Pattern sequences in \(\langle q,r\rangle\)-numeration systems
cites work: Independence results for pattern sequences in distinct bases
cites work: Algebraic independence of the power series defined by blocks of digits
DOI: 10.1007/S10474-011-0082-6
description: scientific article; zbMATH DE number 6006426