The p-adic order of some fibonomial coefficients whose entries are powers of p

Let (F _n ) _n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as

$${left[ {begin{array}{*{20}{c}} m k end{array}} right]_F} = frac{{{F_{m - k + 1}} cdots {F_{m - 1}}{F_m}}}{{{F_1} cdots {F_k}}}$$

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±2 (mod 5), then (p{left| {left[ {begin{array}{*{20}{c}} {{p^{a + 1}}} {{p^a}} end{array}} right]} right._F}) for all integers a ≥ 1. In 2015, Marques and Trojovský worked on the p-adic order of ({left[ {begin{array}{*{20}{c}} {{p^{a + 1}}} {{p^a}} end{array}} right]_F}) for all a ≥ 1 when p ≠ 5. In this paper, we shall provide the exact p-adic order of ({left[ {begin{array}{*{20}{c}} {{p^{a + 1}}} {{p^a}} end{array}} right]_F}) for all integers a, b ≥ 1 and for all prime number p.