İki değişkenli Balans ve Lucas-Balans polinomlarının üreteç matrisleri

Yıl 2021, Cilt: 11 Sayı: 3, 761 - 767, 15.07.2021

Nazmiye Yılmaz

https://doi.org/10.17714/gumusfenbil.841087

Öz

Makalenin amacı iki değişkenli Balans ve Lucas-Balans polinomlarını üçgensel matrislerin determinantları ile ifade etmektir. Ek olarak bu üçgensel matrislerin terslerini elde ettik. Determinantları iki değişkenli Balans ve Lucas-Balans polinomlarının herhangi pozitif ve negatif indisli lineer alt dizilerini üreten üçgensel matrislerin ailesini veren genel sonuçlar ile sonlandırdık.

Anahtar Kelimeler

Balans polinomları, Determinant, Matris tersi, Lucas-balans polinomları, Üçgensel matris

The generating matrices of the bivariate Balancing and Lucas-Balancing polynomials

Öz

The objective of this paper is to express the bivariate Balancing and Lucas-Balancing polynomials in terms of determinants of tridiagonal matrices. In addition, we obtained the inverses of the tridiagonal matrices. We finalized the general results to construct families of the tridiagonal matrices whose determinants generate arbitrary linear subsequence with positive and negative indices of the bivariate Balancing and Lucas-Balancing polynomials.

Anahtar Kelimeler

Balancing polynomials, Determinant, Inverse of matrix, Lucas-Balancing polynomials, Tridiagonal matrix

Kaynakça

• Aşcı, M. and Yakar, M. (2020). Bivariate Balancing polynomials. JP Journal Algebra Number Theory and Applications, 46(1), 97-108. http://dx.doi.org/10.17654/NT046010097
• Behera, A. and Panda, G.K. (1999). On the square roots of triangular numbers. The Fibonacci Quarterly, 37(2), 98-105.
• Cahill, N.D. and Narayan, D.A. (2004). Fibonacci and Lucas numbers as tridiagonal matrix determinants. The Fibonacci Quarterly, 42(3), 216-221.
• Chen, K.W. (2020). Horadam sequences and tridiagonal determinants. Symmetry, 12, 1968. https://doi.org/10.3390/sym12121968
• Falcon, S. (2013). On the generating matrices of the -Fibonacci numbers. Proyecciones Journal of Mathematics, 32(4), 347-357. https://doi.org/10.4067/S0716-09172013000400004
• Feng, J. (2011). Fibonacci identities via the determinant of tridiagonal matrix. Applied Mathematics and Computation, 217(12), 5978-5981. https://doi.org/10.1016/j.amc.2010.12.025
• Frontczak, R. (2019). On Balancing polynomials. Applied Mathematical Sciences, 13(2), 57-66. https://doi.org/10.12988/ams.2019.812183
• Goy, T. (2018). Horadam sequence through recurrent determinants of tridiagonal matrices. Kragujevac Journal of Mathematics, 42(4), 527-532.
• Nalli, A. and Civciv, H. (2009). A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers. Chaos, Solitons and Fractals, 40, 355-361. https://doi.org/10.1016/j.chaos.2007.07.069
• Ozkoc, A. (2015). Tridiagonal matrices via -Balancing number. British Journal of Mathematics and Computer Science, 10(4), 1-11. https://doi.org/10.9734/BJMCS/2015/19014
• Ozkoc, A. and Tekcan, A. (2017). On k-Balancing numbers. Notes on Number Theory and Discrete Mathematics, 23(3), 38-52.
• Panda, G.K. (2009). Some fascinating properties of Balancing numbers. Proceeding of the Eleventh International Conference on Fibonacci Numbers and Their Applications, Congr. Numer. 194, 185-189.
• Patel, B. K., Irmak, N. and Ray, P. K. (2018). Incomplete Balancing and Lucas-Balancing numbers. Mathematical Reports, 20(70), 59-72.
• Ray, P. K. (2012). Application of Chybeshev polynomials in factorizations of Balancing and Lucas-Balancing numbers. Boletim da Sociedade Paranaense de Matematica, 30(2), 49-56. https://doi.org/10.5269/bspm.v30i2.12714
• Ray, P. K. (2017). Balancing polynomials and their derivatives. Ukrainian Mathematical Journal, 69(4), 646-663. https://doi.org/10.1007/s11253-017-1386-7
• Ray, P. K. (2018). On the properties of -Balancing numbers. Ain Shams Engineering Journal, 9, 395-402. https://doi.org/10.1016/j.asej.2016.01.014
• Ray, P. K. and Panda, G. K. (2015). Tridiagonal matrices related to subsequences of Balancing and Lucas-Balancing numbers. Notes on Number Theory and Discrete Mathematics, 21(3), 56-63.
• Strang, G. (1998). Introduction to linear algebra. 2nd ed., Wellesley (MA), Wellesley-Cambridge. Strang, G. and Borre, K. (1997). Linear algebra, geodesy and GPS. Wellesley (MA), Wellesley-Cambridge, 555-7.
• Taskara, N., Uslu, K., Yazlik, Y. and Yilmaz, N. (2011). The construction of Horadam numbers in terms of the determinant of tridiagonal matrices. AIP Conference Proceedings, 1389(1), 367-370. https://doi.org/10.1063/1.3636739
• Trojovsky, P. (2016). On determinants of tridiagonal matrices with -diagonal or superdiagonal in relation to Fibonacci numbers. Global Journal of Pure and Applied Mathematics, 12(2), 1885-1892.
• Usmani, R.A. (1994a). Inversion of a tridiagonal Jacobi matrix. Linear Algebra and its Applications, 212, 413-414. https://doi.org/10.1016/0024-3795(94)90414-6
• Usmani, R.A. (1994b). Inversion of Jacobi’s tridiagonal matrix. Computers & Mathematics with Applications,27, 59-66.
• Yakar, M. (2020). Bivariate Balancing polynomials. MS Thesis, Pamukkale University Institue of Science, Denizli.
• Yilmaz, F. and Kirklar, E. (2015). A note on k-tridiagonal matrices with the Balancing and Lucas-Balancing numbers. Ars Combinatoria, 120, 283-291.
• Yilmaz, N. (2020). Binomial transforms of the Balancing and Lucas-Balancing polynomials. Contributions to the Discrete Mathematics, 15(3), 133-144. https://doi.org/10.11575/cdm.v15i3.69846