Pattern sequences in ‹q, r›- numeration systems

Let q ≥ 2 and 0 ≤ r ≤ q − 2 be integers. In this paper, we study pattern sequences for patterns in ‹q, r›-numeration systems through their generating functions. Our result implies that any nontrivial linear combination over ? of pattern sequences chosen from different ‹q, r›-numeration systems cannot be a linear recurrence sequence. In particular, pattern sequences in different ‹q, r›-numeration systems are linearly independent over ?, while within one ‹q, r›-numeration system they can be linearly dependent ?.