Explicit formulas for the $p$-adic valuations of Fibonomial coefficients II

In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients $\binom{p^a n}{n}_F$ 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,...

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Bibliographic Details
Published inAIMS mathematics Vol. 5; no. 6; pp. 5685 - 5699
Main Authors Phunphayap, Phakhinkon, Pongsriiam, Prapanpong
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2020
Subjects
binomial coefficient
divisibility
fibonacci number
fibonomial coefficient
p$-adic order
p$-adic valuation
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ISSN2473-6988
2473-6988
DOI10.3934/math.2020364

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Summary:In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients $\binom{p^a n}{n}_F$ 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 $\binom{pn}{n}_F$ is divisible by $p$, where $p$ is any prime which is congruent to $\pm 2 \pmod{5}$.
ISSN:2473-6988
2473-6988
DOI:10.3934/math.2020364