A common generalization of dickson polynomials, fibonacci polynomials, and lucas polynomials and applications | [Journal of Prime Research in Mathematics • 2024]

Author(s):

1. Said Zriaa: Department of Mathematics, University Chouaib Doukkali,Eljadida,Morocco

2. Mohammed Mouco¸uf: Department of Mathematics, University Chouaib Doukkali,Eljadida,Morocco

Abstract:

In this work, we define a more general family of polynomials in several variables satisfying a linear recurrence relation. We provide explicit formulas and determinantal expressions. Our results are then applied to secondorder recurrent polynomials, presenting several relationships and identities involving Fibonacci polynomials of order 2, Lucas polynomials of order 2, classical Fibonacci polynomials, classical Lucas polynomials, Fibonacci numbers, Lucas numbers, and both kinds of Dickson polynomials. Our findings ofer a unified generalization of various existing works, with several well-known results emerging as special cases.

Page(s): 30-47

DOI: DOI not available

Published: Journal: Journal of Prime Research in Mathematics, Volume: 20, Issue: 2, Year: 2024

Keywords:

2010 MSC , Lucas numbers , 11B39 , Fibonacci numbers , Dickson polynomials , Fibonacci polynomials , Lucas polynomials , 11B83

References:

[1] Armstrong H. . .A treatment of cascaded active four-terminal networks, with application to transistor circuits. IRE Transactions on Circuit Theory, 3(1956) : 4-140.

[2] Armstrong H. . .Note on the use of Tchebychef functions in dealing with iterated networks. IRE Trans. on Circuit Theory, 2(1955) : 4-70.

[3] Barakat R.,Baumann E. .1969 .Mth Powers of an N × N matrix and its connection with the generalized Lucas polynomials. J. Math. Phys. 10, : 4-1476.

[4] Catarino P. .2018 .Diagonal functions of k-Pell and k-Pell-Lucas polynomials and some identities. Acta Math. Univ. Comenianae, 87 : 1-159.

[5] Catarino P. .2017 .A note on certain matrices with h(x)-Fibonacci polynomials. Acta Math. Univ. Comenianae, 86 : 1-270.

[6] Chen L.,X. Wang. L. .2019 .The power sums involving Fibonacci polynomials and their applications. Symmetry, 635 : 1.

[7] Everest G.,Poorten A. Van Der,Ward T. .2003 .. American Mathematical Society, Mathematical Surveys and Monographs 104, : .

[8] Guan H,Khan W. A.,Ryoo C. S.,Fibonacci Polynomials .2024 .On Certain Properties of Parametric Kinds of Apostol-Type Frobenius-Euler-. , 348 : 1.

[9] Koshy T. .2001 .Fibonacci and Lucas numbers with applications. , : .

[10] Koshy T. .2014 .Pell and Pell-Lucas numbers with applications. , : .

[11] Kuloglu B.,E. B. .2023 .O¨zkan and M. Marin. Fibonacci and Lucas polynomials in n-gon. , 31 : 1-140.

[12] Levesque C. .1985 .On m-th order linear recurrences, Fibonacci Q. , 23 : 3-293.

[13] E. .2019 .O¨zkan, I˙. Altun. Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials. Commun. Algebra, 47(4) : 3-4030.

[14] E. .2021 .O¨zkan, B. Kulgol˘u. On the new Narayana polynomials, the Gauss Narayana numbers and their polynomials. Asian-Eur. J. Math. 14, 2150100 : 1.

[15] Mowery V. .1964 .On hypergeometric functions in iterated networks. IEEE Trans. Circuit Theory, 11 : 4-247.

[16] Mullen G. L.,Lidl R.,Turnwald. Dickson Polynomials G. .1993 .. , 2(5) : 4.

[17] M. C. .1952 .Pease. The iterated network and its application to diferentiators. Proc. IRE, 40 : 4-711.

[18] J. L. .2014 .Ra mır´ez. On convolved generalized Fibonacci and Lucas polynomials. Appl. Math. Comput, 229 : 1-213.

[19] Ricci P. E. .2021 .A note on Q-matrices and higher order Fibonacci polynomials. , 27 : 1-100.

[20] Ricci P. E.,Srivastava R.,Caratelli D. .2024 .Laguerre-Type Bernoulli and Euler Numbers and Related Fractional Polynomials. , 381 : 4.

[21] Soykan. Generalized Tribonacci Y.,Polynomials Y. .2023 .. Earthline Journal of Mathematical Sciences, 13 : 1-20.

[22] Sylvester J. J. .1909 .. Messenger Maths, 8(1879) : 5-251.

[23] Tast M.,The Gauss K-Fibonacci E. O¨zkan. On,Polynomials E. O¨zkan. On . .. J. Math. Anal. Appl, 9(2021) : 1-130.

[24] Tuglu N.,Kocer E. G.,Stakhov A. .2011 .Bivariate Fibonacci like p-polynomials. Appl. Math. Comput, 217 : 1-10246.

[25] Wang T.,W. P. Zhang. T. .2012 .Some identities involving Fibonacci, Lucas polynomials and their applications. Bull. Math. Soc. Sci. Math. Roumanie, 55 : 1-103.

[26] Ye X.,Z. Zhang. X. .2017 .A common generalization of convolved generalized Fibonacci and Lucas polynomials and its applications. Appl. Math. Comput, 306 : 1-37.

[27] Yi Y.,W. P. Zhang. Y. .2002 .Some identities involving the Fibonacci polynomials. , 40 : 1-318.

[28] Zriaa S.,M. S. .2023 .Mouc¸ouf. Explicit, recurrent, determinantal expressions of the kth power of formal power series and applications to the generalized Bernoulli numbers. arXiv:2305.06072[math.NT], : .

Citations

Downloads

Views