Twin Polynomials and Kernels Matrix

Aziz ATTA^1,

¹Atta Engineering Design Office (Study and Technical Assistance), El Jadida, Morocco

Cite this paper:
Aziz ATTA. Twin Polynomials and Kernels Matrix. Turkish Journal of Analysis and Number Theory. 2020; 8(3):57-69. doi: 10.12691/tjant-8-3-2

Abstract

Polynomials and matrices have played a very important role in the development of different branches of mathematics. Indeed, several mathematicians have introduced classical polynomials very useful for the scientific community such as the Lagrange’s interpolation polynomials, the Chebyshev’s polynomials and the Bernstein’s polynomials [1,2]. Also, there is a strong link between polynomials and matrices through the notions of the determinant, the characteristic polynomial and the minimal polynomial. Similarly, we will introduce in this article two polynomials which we will call twin polynomials as well as a matrix called kernels matrix. Finally, we will present some applications such as the resolution of recurrent sequences with second member and the establishment of several sum formulas.

Keywords:
factorial mean twin polynomials factorial means staircase kernels matrix kernels determinant

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	Kil H. Kwon and Lance L. Littlejohn, Classification of classical orthogonal polynomials, J. Korean Math. Soc. 34 (1997), No. 4, pp. 973-1008.
[2]	Brian George Spencer Doman, The classical orthogonal polynomials, Univerty of Liverpool, November 2005. Worldscientific.
[3]	M. Samuel Fiorini, MATH-F-307 Mathématiques discrètes, Version du 5 octobre 2012, Bruxelles.
[4]	Craig Smorynski, A treatise on the Binomial Theorem, King’s college London, UK.
[5]	Jeffrey R. Chasnov, Fibonacci Numbers and the Golden Ratio, www.math.ust.hk/~machas/fibonacci.pdf, The Hong Kong University of Science and Technology, Hong Kong.
[6]	Yashwant K. Panwar1, Mamta Singh, Certain Properties of Generalized Fibonacci Sequence, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 1, 6-8.
[7]	José Lopez-Bonilla, Sergio Vidal Beltran, Jesus Yalja Montiel, Note on Dirichlet and Féjer kernels, Teoria y Aplicaciones 2007 14(1): 101-104, January 2007.
[8]	V. N. Jha, Study on Hermitian, Skew-Hermitian and Uunitary Matrices as a part of Normal Matrices, International Journal of Open Information Technologies ISSN: 2307-8162 vol. 4, no. 11, 2016.
[9]	Jean-Louis Rouget, “Polynômes de Tchebychev,” www.maths-france.fr. 2008.