Complexity of powers of a constant-recursive sequence

TL;DR

The paper analyzes the complexity of powers of constant-recursive sequences by introducing the rank of a sequence and deriving a refined upper bound for the rank of $s(n)^M$ that accounts for root multiplicities: if $s(n)$ has rank $r$ and its characteristic polynomial has $k$ distinct roots, then $\operatorname{rank}(s^M) \le (r-k)\binom{M+k-1}{M-1} + \binom{M+k-1}{M}$. It shows that for Fibonacci numbers this bound is attained, while other sequences may exhibit smaller ranks due to root-product coincidences and coefficient cancellations. The work also initiates the study of rank sequences $\big(\operatorname{rank}(s(n)^M)\big)_{M\ge 1}$, proving a complete classification for rank $2$ (five general sequences) and providing computational data for higher ranks, along with conjectures about the eventual form of general versus particular rank sequences. A symbolic-root framework and related lattice-theoretic ideas are proposed as a path forward to characterize all possible rank sequences, with attention to multiplicities and algebraic constraints on roots.

Abstract

Constant-recursive sequences are those which satisfy a linear recurrence, so that later terms can be obtained as a linear combination of the previous ones. The rank of a constant-recursive sequence is the minimal number of previous terms required for such a recurrence. For a constant-recursive sequence $s(n)$, we study the sequence $\left(\text{rank}\, s(n)^M\right)_{M\geq 1}$. We answer a question of Stinchcombe regarding the complexity of the powers of a constant-recursive sequence when the roots of the characteristic polynomial are not all distinct.

Theorems & Definitions (25)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• proof
• Proposition 2.4
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• Theorem 3.3