Algebraic independence results related to pattern sequences in distinct $\langle q,r \rangle$-numeration systems

Abstract

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.