## Tohoku Mathematical Journal 2012 September SECOND SERIES VOL. 64, NO. 3

 Tohoku Math. J. 64 (2012), 427-438

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

Author Yohei Tachiya

(Received June 1, 2011, revised November 17, 2011)
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.

2000 Mathematics Subject Classification. Primary 11J85; Secondary 11A63, 11J72.

Key words and phrases. Mahler type functional equation, algebraic independence, pattern sequences, $\langle q,r \rangle$-numeration systems.