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

Abstract

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

$${\left[\begin{array}{c}m\\ k\end{array}\right]}_F=\frac{F_{m-k+1}...F_{m-1}{F}_m}{F_1...F_k}$$

and ${\left[\begin{array}{c}m\\ k\end{array}\right]}_F=0$ 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 ${\left[\begin{array}{c}{p}^{a+n}\\ p^a\end{array}\right]}_F$. Furthermore, we also find the conditions of $p$ when ${\left[\begin{array}{c}{p}^{a+n}\\ p^a\end{array}\right]}_F$ is not divisible by $p$.