On the linear space of the two-sided generalized Fibonacci sequences

Abstract: In this paper, we study the linear space of all two-sided generalized Fibonacci sequences ${F_n}{n \in \mathbb{Z}}$ that satisfy the recurrence equation of order $k$: $F_n = F{n-1} + F_{n-2} + \dots + F_{n-k}$. We give two types of explicit formula, one is based on generalized binomial coefficients and the other based on generalized multinomial coefficients.