ON SUMS OF BIVARIATE FIBONACCI POLYNOMIALS AND BIVARIATE LUCAS POLYNOMIALS
Keywords:
Bivariate Fibonacci Polynomials, Bivariate Lucas Polynomials, Binet’s formula and two cross two matrix.
Abstract
In this paper, we present the sum of s+1 consecutive member of Bivariate Fibonacci Polynomials and Bivariate Lucas Polynomials and related identities consisting even and odd terms. We present its two cross two matrix and find interesting properties such as nth power of the matrix. Also, we present the identity which generalizes Catlan’s, Cassini’s and d’Ocagne’s identity. Binet’s formula will employ to obtain the identities.
References
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polynomials and generalized k-Lucas polynomials, Notes on Number Theory
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Bivariate Fibonacci Polynomials and Bivariate Lucas Polynomials, Journal
of Amasya University the Institute of Sciences and Technology, 1 (2) (2020),
146-154.
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(p,q)-modified Pell numbers with Gaussian numbers and polynomials, 28 (4)
(2021), 1311-1327.
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Applied Mathematics and Computation, 217 (24) (2011), 10239-10246.
Help, Acta Didactica Napocensia, 9 (4) (2016), 71-95.
[2] Belbachir, H., and Bencherif, F., On some properties on bivariate Fibonacci
and Lucas polynomials, (2007). arXiv:0710.1451v1.
[3] Cakmak, T., and Karaduman, E., On the derivatives of bivariate Fibonacci
polynomials, Notes on Number Theory and Discrete Mathematics, 24 (3)
(2018), 37-46.
[4] Catalani, M., Generalized Bivariate Fibonacci Polynomials, Version 2 (2004).
http://front.math.ucdavis.edu/math.CO/0211366
[5] Catalani, M., Identities for Fibonacci and Lucas Polynomials derived from a
book of Gould, (2004). http://front.math.ucdavis.edu/math.CO/0407105
[6] Catalani, M., Some Formulae for Bivariate Fibonacci and Lucas Polynomials,
(2004). http://front.math.ucdavis.edu/math.CO/0406323
[7] Cerin, Z., Alternating sums of Lucas numbers, Central European Journal of ˇ
Mathematics, 3 (1) (2005), 1-13.
[8] Cerin, Z., Sums of Product of Generalized Fibonacci and Lucas Numbers, ˇ
Demonstratio Mathematica, 42 (2) (2009), 247-258.
[9] Cerin, Z., On sums of squares of odd and even terms of the Lucas sequence, ˇ
Proceedings of the 11th Fibonacci Conference, (to appear).
[10] Cerin, Z., Properties of odd and even terms of the Fibonacci sequence, ˇ
Demonstratio Mathematica, 39 (1) (2006), 55–60.
[11] Hoggatt, V. E. Jr. and Bicknell, M., Generalized Fibonacci Polynomials, The
Fibonacci Quarterly, 11 (5) (1973), 457-465.
[12] Inoue, K., and Aki, S., Bivariate Fibonacci polynomials of order k with statistical applications, Annals of the Institute of Statistical Mathematics, 63
(1) (2011), 197-210.
[13] Jacob, G., Reutenauer, C., and Sakarovitch, J., On a divisibility property of
Fibonacci polynomials, (2006).
http://perso.telecom-paristech.fr/ jsaka/PUB/Files/DPFP.pdf.
[14] Kaygisiz, K., and Sahin, A., Determinantal and permanental representation
of generalized bivariate Fibonacci p-polynomials, (2011). arXiv:1111.4071v1.
[15] Kim, T., Kim, D. S., Dolgy, D. V. and Park, J. -W., Sums of finite products
of Chebyshev polynomials of the second kind and of Fibonacci polynomials,
J. Inequal. Appl. 2018, 148, (2018).
[16] Kim, D., Dolgy, D. V., Kim, D. S. and Seo, J. J., convolved Fibonacci numbers and their applications, Ars Combin., 135 (2017), 119–131.
[17] Ozkan, E. and Altun, I., Generalized Lucas polynomials and relationships ¨
between the Fibonacci polynomials and Lucas polynomials, Communications
in Algebra, 47 (10) (2019), 4020-4030.
[18] Ozkan, E. and Ta¸stan, M., A new families of Gauss k-Jacobsthal numbers ¨
and Gauss k-Jacobsthal-Lucas numbers and Their Polynomials, Journal of
Science and Arts, 53 (4) (2020), 893-908.
[19] Ozkan, E., and Ta¸stan, M., On a new family of gauss k-Lucas numbers and ¨
their polynomials, Asian-European Journal of Mathematics, 14 (6) (2021),
215010.
[20] Ozkan, E., and Ta¸stan, M., On Gauss Fibonacci polynomials, on Gauss Lu- ¨
cas polynomials and their applications, Communications in Algebra, 48 (3)
(2020), 952-960.
[21] Ta¸stan, M., Ozkan, E. and Shannon, A. G., The generalized k-Fibonacci ¨
polynomials and generalized k-Lucas polynomials, Notes on Number Theory
and Discrete Mathematics, 27 (2) (2021), 148-158.
[22] Panwar, Y. K., and Singh, M., Generalized bivariate Fibonacci-Like polynomials, International journal of Pure Mathematics, 1 (2014), 8-13.
[23] Panwar, Y. K., Gupta, V. K. and Bhandari, J., Generalized Identities of
Bivariate Fibonacci Polynomials and Bivariate Lucas Polynomials, Journal
of Amasya University the Institute of Sciences and Technology, 1 (2) (2020),
146-154.
[24] Saba, N. and Boussayoud, A., Generating functions of binary products of
(p,q)-modified Pell numbers with Gaussian numbers and polynomials, 28 (4)
(2021), 1311-1327.
[25] Tasci, D., Cetin Firengiz, M. and Tuglu, N., Incomplete Bivariate Fibonacci
and Lucas p− Polynomials, Discrete Dynamics in Nature and Society,
doi:10.1155/2012/840345.
[26] Tuglu, N., Kocer, E. G. and Stakhov, A., Bivariate fibonacci like p-polynomials,
Applied Mathematics and Computation, 217 (24) (2011), 10239-10246.
Published
2022-09-24
Section
Articles