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

2024/10/10 by Mane, S. R. · 1 citation
#11B37 #11B39 #39A06 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.07922

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) +…+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≫1. 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.

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