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

The aim of this paper consists of providing summation formulas for the $k$-Fibonacci numbers ($k\in \mathbb{Z}$, $k\ge 2$) 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.