Asymptotics for partitions over the Fibonacci numbers and related sequences

TL;DR

The paper determines precise asymptotics for the number $p_F(n)$ of non-distinct partitions of $n$ over Fibonacci numbers, revealing a dominant non-oscillatory growth tempered by log-periodic fluctuations. The authors derive an exact generating-function framework $F_2(x)=\sum p_F(n)x^n=\prod_{k\ge2}(1-x^{F_k})^{-1}$ and analyze it via Mellin inversion, leveraging the fibonaccian Dirichlet series $\zeta_F(z)$ to obtain a detailed pole structure and an explicit expansion for $\log F_2(e^{-s})$ that drives the saddle-point analysis. Using the Coons–Kirsten saddle-point method, they obtain a refined asymptotic for $p_F(n)$ of the form $p_F(n)=\psi_2(n)\,n^{a(n)}(\log n)^{b(n)}(1+O((\log\log n)^2/\log n))$, with $a(n),b(n)$ and the periodic amplitude $\psi_2$ encoding oscillations via $1$-periodic functions. The framework further generalizes to linear recurrences with a dominant root $\beta$, yielding analogous asymptotics $p_P(n)=A_P\left(\frac{\log n}{\log\beta}\right)n^{B_P(n)}(\log n)^{C_P(n)}(1+O((\log\log n)^2/\log n))$, thus broadening the scope of oscillatory partition asymptotics beyond Fibonacci numbers.

Abstract

In this paper, harkening back to ideas of Hardy and Ramanujan, Mahler and de Bruijn, with the addition of more recent results on the Fibonacci Dirichlet series, we determine the asymptotic number of ways $p_F(n)$ to write an integer as the sum of non-distinct Fibonacci numbers. This appears to be the first such asymptotic result concerning non-distinct partitions over Fibonacci numbers. As well, under weak conditions, we prove analogous results for a general linear recurrences.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1: Navas, 2001
• Lemma 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof