On some new identities for the Fibonomial coefficients

Abstract

Let F _n be the nth Fibonacci number. The Fibonomial coefficients $\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F$ are defined for n ≥ k > 0 as follows

$$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$

with $\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1$ and $\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0$. In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that

$$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$

holds for all non-negative integers n and l.