On a Congruence Involving Generalized Fibonomial Coefficients.

Let ( F ) be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as . 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 . 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. [ABSTRACT FROM AUTHOR]