Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients II

Abstract

In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients ${\left(\genfrac{}{}{0ex}{}{p^{a}n}{n}\right)}_F$ for all primes $p$ and positive integers $a$ and $n$. This is a continuation from our previous article extending some results in the literature, which deal only with $p=2,3,5,7$ and $a=1$. Then we use these formulas to characterize the positive integers $n$ such that ${\left(\genfrac{}{}{0ex}{}{pn}{n}\right)}_F$ is divisible by $p$, where $p$ is any prime which is congruent to $\pm 2(\mathrm{mod}5)$.

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