Algebraic independence results related to pattern sequences in distinct \(\langle q,r \rangle\)-numeration systems
Let \(q\geq 2\) and \(0\leq r\leq q-2\) be integers. In this paper, the author gives some algebraic independence results of the generating functions of pattern sequences for words in \(\langle q,r\rangle\)--numeration systems. This is done by showing that the generating functions of such systems satisfy certain Mahler--type functional equations. Recall that in the \(\langle q,r\rangle\)--numeration system (a generalization of the standard \(q\)--ary numeration), every positive integer \(n\) is uniquely expressed as \[ n=\sum_{i=0}^ka_i q^i,\quad a_i\in\Sigma_{q,r},\quad a_k>0, \] where \[ \Sigma_{q,r}:=\{-r,1-r,\ldots,0,1,\ldots,q-1-r\}. \] Denote the set of all (finite) words over \(\Sigma_{q,r}\) by \(\Sigma_{q,r}^*\), and define \(e_{q,r}(w;n)\) to be the number of (possibly overlapping) occurrences of \(w\) in the \(\langle q,r\rangle\)--expansion of \(n\). The sequence \[ \{e_{q,r}(w;n)\}_{n\geq 0} \] is called the \textit{pattern sequence} for the word \(w\) in the \(\langle q,r\rangle\)--numeration system. The authors' main result is the following. Let \(w_{q,r}\in\Sigma_{q,r}^*\) for \(0\leq r\leq q-2)\) and \(q=2,3,\ldots,\) be any nonzero patterns. Then the functions \[ f_{q,r}(z)=\sum_{n\geq 0}e_{q,r}(w_{q,r};n)z^n\quad (0\leq r\leq q-2, q=2,3,\ldots) \] are algebraically independent over \(\mathbb{C}(z)\). Moreover, the values \(f_{q,r}(\alpha)\), for \(q\) and \(r\) in the ranges denoted above, are algebraically independent over \(\mathbb{Q}\) for any algebraic number \(\alpha\) with \(0<|\alpha|<1\). These generalise results of \textit{I.~Shiokawa} and the author [Indag. Math., New Ser. 19, No. 1, 151--161 (2008; Zbl 1185.11010); Acta Math. Hung. 132, No. 1--2, 190--206 (2011; Zbl 1249.11014)], as well as a previous result of the author [Acta Arith. 142, No. 2, 157--167 (2010; Zbl 1220.11010)].
\(\langle q,r \rangle\)-numeration systems
scientific article; zbMATH DE number 6110245