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

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 $k$: $F_n=F_{n-1}+F_{n-2}+\cdots +F_{n-k}$. We give two types of explicit formula, one is based on generalized binomial coefficients and the other based on generalized multinomial coefficients.