p-nomial Coefficients and p-nomial Theorem

Aziz ATTA^1,

¹Mathematics and Structural Analysis, Atta Engineering Design Office, El Jadida, Morocco

Cite this paper:
Aziz ATTA. p-nomial Coefficients and p-nomial Theorem. Turkish Journal of Analysis and Number Theory. 2020; 8(1):6-15. doi: 10.12691/tjant-8-1-2

Abstract

Binomial coefficients have long been studied by several mathematicians for several centuries [1] and they are currently grouped in what is called Pascal's triangle [2]. These coefficients are very useful not only in combinatorics, but also, they intervene in many fields such as enumeration, development of the binomial in algebra, development in series, and in probability distributions and statistics [3]. In addition, Pascal's formula, generative of these coefficients, leads us to think of a generalization of these coefficients and of the other strongly related mathematical tools. In this article, we will try to generalize the binomial coefficients, the Newton’s binomial as well as the Fibonacci sequence and establish their expressions. We will call these generalizations p-nomial coefficients, Atta’s p-nomial and p-bonacci sequence.

Keywords:
p-nomial coefficients p-nomial identity staircase of p-nomials p-bonacci sequence p-Bernstein polynomial

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References:

[1]	Carl Benjamin Boyer, A History of Mathematics, New York, 2nd, 1991. Available: Amazon.
[2]	M. Samuel Fiorini, MATH-F-307 Mathématiques discrètes, Version du 5 octobre 2012, Bruxelles.
[3]	Richard A. Brualdi, Introductory Combinatorics -Fifth Edition-, (ISBN 978-0-13-602040-0) [2009].
[4]	Paolo Emilio Ricci, A Note on Golden Ratio and Higher Order Fibonacci Sequences, Turkish Journal of Analysis and Number Theory, 2020.
[5]	Serkan Araci, Mehmet Acikgoz, Armen Bagdasaryan, Erdoğan Şen, The Legendre Polynomials Associated with Bernoulli, Euler, Hermite and Bernstein Polynomials, 2013.
[6]	M. Roberto Corcino, On p,q-binomial coefficients, Electronic Journal of Combinatorial Number Theory 8 (2008).
[7]	Bijendra Singh, Omprakash Sikhwal and Yogesh Kumar Gupta, Generalized Fibonacci-Lucas Sequence, Turkish Journal of Analysis and Number Theory, 2014.
[8]	M. José Luis Ramírez and M. Victor F. Sirvent, A Generalization of the k-Bonacci Sequence from Riordan Arrays, The electronic journal of combinatorics· February 2015.
[9]	Yüksel Soykan, İnci Okumuş, On a generalized Tribonacci sequence, Journal of Progressive Research in Mathematics (JPRM), January 2019.
[10]	Giuseppe Modica and Laura Poggiolini, A First Course in Probability and Markov Chains, University of Firenze, Italy. Available: pdfdrive.
[11]	Craig Smorynski, A treatise on the Binomial Theorem, King’s college London, UK. Available: pdfdrive.
[12]	European Centre for Research Training and Development UK, Steps problem: the link between combinatoric and k-bonacci sequences, European Journal of Statistics and Probability, Vol.3, No.4, pp.10-19, December 2015.