We introduce generalized involutive operators on Banach spaces, based on the notion of a spectral family.

The $k$-Fibonacci sequence $\{g_n^{(k)}\}$ is defined as: $$ g_1^{(k)}=\ldots=g_{k-2}^{(k)}=0,\;\;g_{k-1}^{(k)}=g_{k}^{(k)}=1 $$ and for $n>k\ge 2$, $$ g_n^{(k)}=g_{n-1}^{(k)}+g_{n-2}^{(k)}+\cdots+g_{n-k}^{(k)}. $$ In this paper, we give a combinatorial representation of $g_n^{(k)}$ and give some properties for $k$-Fibonacci sequence.