Asymptotic Analysis of q-Recursive Sequences

Abstract

For an integer $$q\ge 2$$ q ≥ 2 , a q-recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of q. In this article, q-recursive sequences are studied and the asymptotic behavior of their summatory functions is analyzed. It is shown that every q-recursive sequence is q-regular in the sense of Allouche and Shallit and that a q-linear representation of the sequence can be computed easily by using the coefficients from the recurrence relations. Detailed asymptotic results for q-recursive sequences are then obtained based on a general result on the asymptotic analysis of q-regular sequences.Three particular sequences are studied in detail: We discuss the asymptotic behavior of the summatory functions of Stern’s diatomic sequence, the number of non-zero elements in some generalized Pascal’s triangle and the number of unbordered factors in the Thue–Morse sequence. For the first two sequences, our analysis even leads to precise formulæ without error terms.

Decidability and k-regular sequences

“…The authors of [7] proved that every k-recursive sequence is k-regular. This suggests the question of whether there is a k-regular sequence that is not k-recursive.…”

“…As every strongly k-recursive sequence is k-recursive, we conclude that every k-automatic sequence is also k-recursive. The converse, however, is not true, as there are k-recursive and strongly k-recursive sequences that are not bounded (see [7]), but automatic sequences are bounded (see [2]).…”

“…More about synchronized sequences can be found, for example, in [9].It is known that every k-automatic sequence is k-synchronized (see [9, Theorem 4]), and every k-synchronized sequence is k-regular (see [9, Theorem 6]).Recently Heuberger, Krenn, and Lipnik [7] introduced a fourth related class of sequences, the k-recursive sequences.4. We say a sequence (f (n)) n≥0 is k-recursive if there exist two natural numbers r < t and two integers L < U and a natural number n 0 such that every subsequence of the form (f (k t n + b)) n≥n 0 with 0 ≤ b < k t is a linear combination of the elements of the setThe authors of [7] proved that every k-recursive sequence is k-regular. This suggests the question of whether there is a k-regular sequence that is not k-recursive.…”

“…In this paper, we provide an affirmative answer (Theorem 9).Strongly k-recursive sequences. The investigation of the question above leads to a natural variation on the definition in [7] and studying the corresponding class of sequences: in the definition of a k-recursive sequence, we insist that n 0 = 0, L ≥ 0, and U ≤ k t . We call such a sequence strongly k-recursive.Clearly, every strongly k-recursive sequence is k-recursive.…”