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Asymptotic Analysis of q-Recursive Sequences

Clemens Heuberger, Daniel Krenn, Gabriel F. Lipnik

Abstract

For an integer $q\ge2$, 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.

Asymptotic Analysis of q-Recursive Sequences

Abstract

For an integer , a -recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of~. In this article, -recursive sequences are studied and the asymptotic behavior of their summatory functions is analyzed. It is shown that every -recursive sequence is -regular in the sense of Allouche and Shallit and that a -linear representation of the sequence can be computed easily by using the coefficients from the recurrence relations. Detailed asymptotic results for -recursive sequences are then obtained based on a general result on the asymptotic analysis of -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.

Paper Structure

This paper contains 46 sections, 25 theorems, 131 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction
  2. $q$-Recursive Sequences.
  3. $q$-Regular Sequences.
  4. Linear Representation of $q$-Recursive Sequences.
  5. Asymptotics of $q$-Recursive Sequences.
  6. Explicit Precise Asymptotics for Three Particular Sequences.
  7. Proofs.
  8. Brief Introduction to $q$-Regular Sequences
  9. $q$-Recursive Sequences
  10. Definitions
  11. Reduction to $q$-Regular Sequences in the General Case
  12. Reduction to $q$-Regular Sequences in a Special Case
  13. Asymptotics
  14. Growth of Matrix Products
  15. Asymptotics for Regular Sequences
  16. ...and 31 more sections

Key Result

Theorem 1

Let $x$ be a $q$-recursive sequence with offset $n_{0}$, exponents $M$ and $m$ and index shift bounds $\ell$ and $u$. Furthermore, setWe use Iverson's convention: For a statement $S$, we set $\llbracket S\rrbracket = 1$ if $S$ is true and $0$ otherwise; see also Graham, Knuth and Patashnik Graham-Kn Then $x$ is $q$-regular with offset $n_{1} = n_{0} - \lfloor\ell' /q^{M}\rfloor$, and a $q$-linear

Figures (4)

  • Figure 5.1: Fluctuation in the main term of the asymptotic expansion of the summatory function $D$. The plot shows the periodic fluctuation $\Phi_{D}(u)$ approximated by its Fourier series of degree $2000$ (red) as well as the function $D(2^{u})/2^{\kappa u}$ (blue).
  • Figure 6.1: Non-zero elements in the generalized Pascal's triangle $\mathcal{P}_2$
  • Figure 6.2: Fluctuation in the asymptotic expansion of the summatory function $Z$. The plot shows the periodic fluctuation $\Phi_{Z}(u) = 2\Phi_{D}(u)$ approximated by its Fourier polynomial of degree $2000$ (red) as well as the function $Z(2^{u})/2^{\kappa u}$ (blue).
  • Figure 7.1: Fluctuation in the main term of the asymptotic expansion of the summatory function $F$. The plot shows the periodic fluctuation $\Phi_{F}(u)$ approximated by its Fourier series of degree $2000$ (red) as well as the function $F(2^{u})/2^{\kappa u}$ (blue).

Theorems & Definitions (61)

  • Example 2.1: Binary Sum of Digits OEIS:2021
  • Definition 3.1: $q$-Recursive Sequence
  • Remark 3.2
  • Example 3.3: Odd Entries in Pascal's Triangle OEIS:2021
  • Definition 3.4: $q$-Regular Sequence with Offset
  • Remark 3.5
  • Theorem 1
  • Remark 3.6
  • Example 3.7: Odd Entries in Pascal's Triangle, continued
  • Example 3.8
  • ...and 51 more