Some applications of the Lagrange inversion formula for the $k$-Fibonacci numbers

TL;DR

This paper derives summation formulas and asymptotic expressions for $k$-Fibonacci numbers using the Lagrange inversion formula and Hermite's consequence, expanding understanding of their combinatorial properties.

Contribution

It introduces new summation and asymptotic formulas for $k$-Fibonacci numbers utilizing the Lagrange inversion formula, a novel approach in this context.

Findings

Derived explicit summation formulas for $k$-Fibonacci numbers.

Established asymptotic equivalents in terms of generalized binomial coefficients.

Enhanced analytical tools for studying $k$-Fibonacci sequences.

Abstract

The aim of this paper consists of providing summation formulas for the $k$-Fibonacci numbers ($k \in \mathbb{Z}$, $k \geq 2$) and their asymptotic equivalents in terms of generalized binomial coefficients. Our main tools are the Lagrange inversion formula and one of its consequences due to Hermite.