ALGEBRAIC INDEPENDENCE RESULTS RELATED TO PATTERN SEQUENCES IN DISTINCT $\langle\lowercase{q}, \lowercase{r}\rangle$-NUMERATION SYSTEMS

抄録

In this paper, we prove the algebraic independence over $\boldsymbol{C}(z)$ of the generating functions of pattern sequences defined in distinct $\langle q, r\rangle$-numeration systems. Our result asserts that any nontrivial linear combination over $\boldsymbol{C}$ of pattern sequences chosen from distinct $\langle q, r \rangle$-numeration systems can not be a linear recurrence sequence. As an application, we give a linear independence over $\boldsymbol{C}$ of the pattern sequences.