On a Congruence Involving Generalized Fibonomial Coefficients

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

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

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±1 (mod 5), then p?({left[ {begin{array}{*{20}{c}} {{p^{a + 1}}} {{p^a}} end{array}} right]_F}) for all integers a ≥ 1. In 2010, in particular, Kilic generalized the Fibonomial coefficients for

$${left[ {begin{array}{*{20}{c}} n k end{array}} right]_{F,m}} = frac{{{F_{left( {n - k + 1} right)m}} cdots {F_{left( {n - 1} right)m}}{F_{nm}}}}{{{F_m} cdots {F_{km}}}}$$

. In this note, we generalize Marques, Sellers and Trojovský result to prove, in particular, that if p ≡ ±1 (mod 5), then ({left[ {begin{array}{*{20}{c}} {{p^{a + 1}}} {{p^a}} end{array}} right]_{F,m}} equiv 1) (mod p), for all a ≥ 0 and m ≥ 1.