A note on the power sums of the number of Fibonacci partitions

TL;DR

This paper investigates the asymptotic behavior of power sums of Fibonacci partition counts, establishing growth rates involving spectral radius concepts and automata theory, extending previous results for all positive integers p.

Contribution

It proves a general asymptotic formula for the power sums of Fibonacci partition counts for all positive p, linking growth to spectral radius and automata theory.

Findings

Established asymptotic growth rate for all positive p

Connected spectral radius to Fibonacci partition sums

Showed limit of spectral radius ratio as p approaches infinity

Abstract

For every nonnegative integer $n$, let $r_F(n)$ be the number of ways to write $n$ as a sum of Fibonacci numbers, where the order of the summands does not matter. Moreover, for all positive integers $p$ and $N$, let \begin{equation*} S_{F}^{(p)}(N) := \sum_{n = 0}^{N - 1} \big(r_F(n)\big)^p . \end{equation*} Chow, Jones, and Slattery determined the order of growth of $S_{F}^{(p)}(N)$ for $p \in \{1,2\}$. We prove that, for all positive integers $p$, there exists a real number $\lambda_p > 1$ such that \begin{equation*} S^{(p)}_F(N) \asymp_p N^{(\log \lambda_p) /\!\log \varphi} \end{equation*} as $N \to +\infty$, where $\varphi := (1 + \sqrt{5})/2$ is the golden ratio. Furthermore, we show that \begin{equation*} \lim_{p \to +\infty} \lambda_p^{1/p} = \varphi^{1/2} . \end{equation*} Our proofs employ automata theory and a result on the generalized spectral radius due to Blondel and Nesterov.