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

Phakhinkon Phunphayap; Prapanpong Pongsriiam; Phakhinkon Phunphayap; Prapanpong Pongsriiam

doi:10.3934/math.2020364

AIMS Mathematics

2020, Volume 5, Issue 6: 5685-5699. doi: 10.3934/math.2020364

Previous Article Next Article

Research article

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

Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom, 73000, Thailand

Received: 20 February 2020 Accepted: 30 June 2020 Published: 06 July 2020
MSC : 11B39; 11B65; 11A63

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 $\pm 2 \pmod{5}$.
- Fibonacci number,
- binomial coefficient,
- Fibonomial coefficient,
- $p$-adic valuation,
- $p$-adic order,
- divisibility
Citation: Phakhinkon Phunphayap, Prapanpong Pongsriiam. Explicit formulas for the $p$-adic valuations of Fibonomial coefficients II[J]. AIMS Mathematics, 2020, 5(6): 5685-5699. doi: 10.3934/math.2020364

Related Papers:

Abstract

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 $\pm 2 \pmod{5}$.

References

[1]	S. Aursukaree, T. Khemaratchatakumthorn, P. Pongsriiam, Generalizations of Hermite's identity and applications, Fibonacci Quart., 57 (2019), 126-133.
[2]	C. Ballot, Divisibility of Fibonomials and Lucasnomials via a general Kummer rule, Fibonacci Quart., 53 (2015), 194-205.
[3]	C. Ballot, The congruence of Wolstenholme for generalized binomial coefficients related to Lucas sequences, J. Integer Seq., 18 (2015), Article 15.5.4.
[4]	C. Ballot, Lucasnomial Fuss-Catalan numbers and related divisibility questions, J. Integer Seq., 21 (2018), Article 18.6.5.
[5]	C. Ballot, Divisibility of the middle Lucasnomial coefficient, Fibonacci Quart., 55 (2017), 297-308.
[6]	F. N. Castro, O. E. González, L. A. Medina, The p-adic valuation of Eulerian numbers: Trees and Bernoulli numbers, Experiment. Math., 2 (2015), 183-195.
[7]	W. Chu, E. Kilic, Quadratic sums of Gaussian q-binomial coefficients and Fibonomial coefficients, Ramanujan J., 51 (2020), 229-243. doi: 10.1007/s11139-018-0023-x
[8]	W. Chu, E. Kilic, Cubic sums of q-binomial coefficients and the Fibonomial coefficients, Rocky Mountain J. Math., 49 (2019), 2557-2569. doi: 10.1216/RMJ-2019-49-8-2557
[9]	Y. L. Feng, M. Qiu. Some results on p-adic valuations of Stirling numbers of the second kind, AIMS Math., 5 (2020), 4168-4196.
[10]	R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Second Edition, Addison-Wesley, 1994.
[11]	V. J. W. Guo, Proof of a generalization of the (B.2) supercongruence of Van Hamme through a q-microscope, Adv. Appl. Math., 116 (2020), Art. 102016.
[12]	V. J. W. Guo, J. C. Liu, Some congruences related to a congruence of Van Hamme, Integral Transforms Spec. Funct., 31 (2020), 221-231. doi: 10.1080/10652469.2019.1685991
[13]	V. J. W. Guo, W. Zudilin, A common q-analogue of two supercongruences, Results Math., 75(2020), Art. 46.
[14]	S. F. Hong, M. Qiu, On the p-adic properties of Stirling numbers of the first kind, Acta Math. Hungar., 161 (2020), 366-395. doi: 10.1007/s10474-020-01037-2
[15]	M. Jaidee, P. Pongsriiam, Arithmetic functions of Fibonacci and Lucas numbers, Fibonacci Quart., 57 (2019), 246-254.
[16]	N. Khaochim, P. Pongsriiam, The general case on the order of appearance of product of consecutive Lucas numbers, Acta Math. Univ. Comenian., 87 (2018), 277-289.
[17]	N. Khaochim, P. Pongsriiam, On the order of appearance of product of Fibonacci numbers, Contrib. Discrete Math., 13 (2018), 45-62.
[18]	E. Kilic, I. Akkus, On Fibonomial sums identities with special sign functions: Analytically q-calculus approach, Math. Slovaca, 68 (2018), 501-512. doi: 10.1515/ms-2017-0120
[19]	E. Kilic, I. Akkus, H. Ohtsuka, Some generalized Fibonomial sums related with the Gaussian q-binomial sums, Bull. Math. Soc. Sci. Math. Roumanie, 55 (2012), 51-61.
[20]	E. Kilic, H. Prodinger, Closed form evaluation of sums containing squares of Fibonomial coefficients, Math. Slovaca, 66 (2016), 757-767.
[21]	E. Kilic, H. Prodinger, Evaluation of sums involving products of Gaussian q-binomial coefficients with applications, Math. Slovaca, 69 (2019), 327-338. doi: 10.1515/ms-2017-0226
[22]	D. Knuth, H. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396 (1989), 212-219.
[23]	T. Komatsu, P. Young, Exact p-adic valuations of Stirling numbers of the first kind, J. Number Theory, 177 (2017), 20-27. doi: 10.1016/j.jnt.2017.01.023
[24]	D. Marques, J. A. Sellers, P. Trojovský, On divisibility properties of certain Fibonomial coefficients by a prime, Fibonacci Quart., 51 (2013), 78-83.
[25]	D. Marques, P. Trojovský, On parity of Fibonomial coefficients, to appear in Util. Math.
[26]	D. Marques, P. Trojovský, On divisibility of Fibonomial coefficients by 3, J. Integer Seq., 15 (2012), Article 12.6.4.
[27]	D. Marques, P. Trojovský, The p-adic order of some Fibonomial coefficients, J. Integer Seq., 18 (2015), Article 15.3.1
[28]	K. Onphaeng, P. Pongsriiam, Subsequences and divisibility by powers of the Fibonacci numbers, Fibonacci Quart., 52 (2014), 163-171.
[29]	K. Onphaeng, P. Pongsriiam, Jacobsthal and Jacobsthal-Lucas numbers and sums introduced by Jacobsthal and Tverberg, J. Integer Seq., 20 (2017), Article 17.3.6.
[30]	K. Onphaeng, P. Pongsriiam, The converse of exact divisibility by powers of the Fibonacci and Lucas numbers, Fibonacci Quart., 56 (2018), 296-302.
[31]	P. Phunphayap, P. Pongsriiam, Explicit formulas for the p-adic valuations of Fibonomial coefficients, J. Integer Seq., 21 (2018), Article 18.3.1.
[32]	P. Pongsriiam, Exact divisibility by powers of the Fibonacci and Lucas number, J. Integer Seq., 17 (2014), Article 14.11.2.
[33]	P. Pongsriiam, A complete formula for the order of appearance of the powers of Lucas numbers, Commun. Korean Math. Soc., 31 (2016), 447-450. doi: 10.4134/CKMS.c150161
[34]	P. Pongsriiam, Factorization of Fibonacci numbers into products of Lucas numbers and related results, JP J. Algebra Number Theory Appl., 38 (2016), 363-372.
[35]	P. Pongsriiam, Fibonacci and Lucas numbers associated with Brocard-Ramanujan equation, Commun. Korean Math. Soc., 32 (2017), 511-522.
[36]	P. Pongsriiam, Fibonacci and Lucas Numbers which are one away from their products, Fibonacci Quart., 55 (2017), 29-40.
[37]	P. Pongsriiam, Fibonacci and Lucas numbers which have exactly three prime factors and some unique properties of F18 and L18, Fibonacci Quart., 57 (2019), 130-144.
[38]	P. Pongsriiam, The order of appearance of factorials in the Fibonacci sequence and certain Diophantine equations, Period. Math. Hungar., 79 (2019), 141-156. doi: 10.1007/s10998-018-0268-6
[39]	B. Prempreesuk, P. Noppakaew, P. Pongsriiam, Zeckendorf representation and multiplicative inverse of Fm mod Fn, Int. J. Math. Comput. Sci., 15(2020), 17-25.
[40]	M. Qiu, S. F. Hong, 2-Adic valuations of Stirling numbers of the first kind, Int. J. Number Theory, 15 (2019), 1827-1855. doi: 10.1142/S1793042119501021
[41]	Z. W. Sun, Fibonacci numbers modulo cubes of prime, Taiwanese J. Math., 17 (2013), 1523-1543. doi: 10.11650/tjm.17.2013.2488
[42]	P. Trojovský, The p-adic order of some Fibonomial coefficients whose entries are powers of p, p-Adic Numbers Ultrametric Anal. Appl., 9 (2017), 228-235. doi: 10.1134/S2070046617030050
[43]	W. Zudilin, Congruences for q-binomial coefficients, Ann. Combin., 23 (2019), 1123-1135. doi: 10.1007/s00026-019-00461-8

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)