Arithmetic properties of generalized Fibonacci sequences

Abstract

The generalized Fibonacci sequences are sequences $\{f_n\}$ which satisfy the recurrence $f_n(s, t) = sf_{n - 1}(s, t) + tf_{n - 2}(s, t)$ ($s, t \in \mathbb{Z}$) with initial conditions $f_0(s, t) = 0$ and $f_1(s, t) = 1$. In a recent paper, Amdeberhan, Chen, Moll, and Sagan considered some arithmetic properites of the generalized Fibonacci sequence. Specifically, they considered the behavior of analogues of the $p$-adic valuation and the Riemann zeta function. In this paper, we resolve some conjectures which they raised relating to these topics. We also consider the rank modulo $n$ in more depth and find an interpretation of the rank in terms of the order of an element in the multiplicative group of a finite field when $n$ is an odd prime. Finally, we study the distribution of the rank over different values of $s$ when $t = -1$ and suggest directions for further study involving the rank modulo prime powers of generalized Fibonacci sequences.