Solutions for $k$-generalized Fibonacci numbers using Fuss-Catalan numbers

TL;DR

This paper derives new explicit formulas for $k$-generalized Fibonacci numbers using Fuss-Catalan numbers, analyzes their roots, and extends previous results with asymptotic and initial condition solutions.

Contribution

It introduces novel explicit root expressions, asymptotic accuracy assessments, and a basis for solving the recurrence with arbitrary initial conditions.

Findings

Explicit Fuss-Catalan based root formulas

Quantified asymptotic approximation accuracy

New identity for companion matrices

Abstract

We present new expressions for the $k$-generalized Fibonacci numbers, say $F_k(n)$. They satisfy the recurrence $F_k(n) = F_k(n-1) +\dots+F_k(n-k)$. Explicit expressions for the roots of the auxiliary (or characteristic) polynomial are presented, using Fuss-Catalan numbers. Properties of the roots are enumerated. We quantify the accuracy of asymptotic approximations for $F_k(n)$ for $n\gg1$. Our results subsume and extend some results published by previous authors. We also present a basis (or `fundamental solutions') to solve the above recurrence for arbitrary initial conditions. We comment on the use of generating functions and multinomial sums for the $k$-generalized Fibonacci numbers and related sequences. We note that the resulting multinomial sums are Dickson polynomials of the second kind in several variables. We also present what may be a new identity for companion matrices.