On the extent of the divisibility of Fibonomial coefficients by a prime number

Abstract

Let $(F_{n})_{n \geq 0}$ be the Fibonacci sequence and $p$ be a prime number. For $1 \leq k \leq m$, the Fibonomial coefficient is defined as

$$\begin{bmatrix} m\\k \end{bmatrix}_{F}= \frac{F_{m-k+1}...F_{m-1}F_{m}}{F_{1}...F_{k}}$$

and $\begin{bmatrix} m\\k \end{bmatrix}_{F}=0$ when $k>m$. Let $a$ and $n$ be positive integers. In this paper, we find the conditions of prime number $p$ which divides Fibonomial coefficient $\begin{bmatrix} p^{a+n}\\p^{a} \end{bmatrix}_{F}$. Furthermore, we also find the conditions of $p$ when $\begin{bmatrix} p^{a+n}\\p^{a} \end{bmatrix}_{F}$ is not divisible by $p$.