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Year 2025, Volume: 13 Issue: 1, 78 - 86, 30.04.2025

Hayrullah Özimamoğlu

Abstract

References

  • [1] F. R. V. Alves, Bivariate Mersenne polynomials and matrices, Notes on Number Theory and Discrete Mathematics, 26(3) (2020), 83-95.
  • [2] Q. Bao and D. Yang, Notes on q-partial differential equations for q-Laguerre polynomials and little q-Jacobi polynomials, Fundamental Journal of Mathematics and Applications, 7(2) (2024), 59-76.
  • [3] M. Bayat and H. Teimoori, The linear algebra of the generalized Pascal functional matrix, 295(1–3) (1999), 81–89.
  • [4] R. Brawer, Potenzen der Pascalmatrix und eine identit¨at der kombinatorik, Elem. Math., 45 (1990), 107-110.
  • [5] R. Brawer and M. Pirovino, The linear algebra of the Pascal matrix, Linear Algebra Appl., 174 (1992), 13-23.
  • [6] G. S. Call and D. J. Velleman, Pascal’s matrices, Amer. Math. Monthly, 100(4) (1993), 372–376.
  • [7] M. Catalani, Generalized bivariate Fibonacci polynomials, arXiv:math/0211366v2 [math.CO], (2004).
  • [8] M. Catalani, Some formulae for bivariate Fibonacci and Lucas polynomials, arXiv:math/0406323v1 [math.CO], (2004).
  • [9] M. Catalani, Identities for Fibonacci and Lucas polynomials derived from a book of Gould, arXiv:math/0407105v1 [math.CO], (2004).
  • [10] G. S. Cheon, H. Kim and L. W. Shapiro, A generalization of Lucas polynomial sequence, Discrete Appl. Math., 157(5) (2009), 920–927.
  • [11] S. Halıcı and Z. Aky¨uz, On some formulae for bivariate Pell polynomials, Far East Journal of Applied Mathematics, 41(2) (2010), 101-110.
  • [12] E. G. Kocer, Bivariate Vieta–Fibonacci and bivariate Vieta–Lucas polynomials, IOSR Journal of Mathematics, 12(4) 2016, 44-50.
  • [13] G. H. Lawden, Pascal matrices, Math. Gaz., 56(398) (1972), 325-327.
  • [14] G. Y. Lee, J. S. Kim and S. H. Cho, Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math., 130 (2003), 527–534.
  • [15] G. Y. Lee and M. Asci, Some properties of the (p;q)-Fibonacci and (p;q)-Lucas polynomials, J. Appl. Math., 2012 (2012), 264842.
  • [16] A. Nalli and P. Haukkanen, On generalized Fibonacci and Lucas polynomials, Chaos Solitons Fractals, 42(5) (2009), 3179–3186.
  • [17] H. Özimamoğlu and A. Kaya, The linear algebra of the Pell-Lucas matrix, Fundamental Journal of Mathematics and Applications, 7(3), (2024), 158-168
  • [18] B. Sadaoui and A. Krelifa, d-Fibonacci and d-Lucas polynomials, J. Math. Model., 9(3) (2021), 425-436.
  • [19] L. W. Shapiro, S. Getu, W. J. Woan and L .C. Woodson, The Riordan group, Discrete Appl. Math., 34(1-3) (1991), 229–239.
  • [20] S. Uygun, Bivariate Jacobsthal and Jacobsthal Lucas polynomial sequences, J. Math. Computer Sci., 21 (2020), 176-185.
  • [21] Z. Zhang and X.Wang, A factorization of the symmetric Pascal matrix involving the Fibonacci matrix, Discrete Appl. Math., 155(17) (2007), 2371–2376.
  • [22] Z. Zhizheng, The linear algebra of the generalized Pascal matrix, Linear Algebra Appl., 250 (1997), 51–60.

On a New Generalization of Bivariate Fibonacci Polynomials and Their Matrix Representations

Year 2025, Volume: 13 Issue: 1, 78 - 86, 30.04.2025

Hayrullah Özimamoğlu

Abstract

In this study, we introduce the concept of d-bivariate Fibonacci polynomials, which generalize the classical bivariate Fibonacci polynomials. We obtain several fundamental properties for these new polynomials including the generating function, the Binet’s formula, combinatorial identities and summation formulas. Then, we define the infinite d-bivariate Fibonacci polynomials matrix, which is a Riordan matrix. By Riordan method, we give two new factorizations of the infinite Pascal matrix including the d-bivariate Fibonacci polynomials.

Keywords

Bivariate Fibonacci polynomials d-bivariate Fibonacci polynomials Pascal matrix Riordan matrix

References

  • [1] F. R. V. Alves, Bivariate Mersenne polynomials and matrices, Notes on Number Theory and Discrete Mathematics, 26(3) (2020), 83-95.
  • [2] Q. Bao and D. Yang, Notes on q-partial differential equations for q-Laguerre polynomials and little q-Jacobi polynomials, Fundamental Journal of Mathematics and Applications, 7(2) (2024), 59-76.
  • [3] M. Bayat and H. Teimoori, The linear algebra of the generalized Pascal functional matrix, 295(1–3) (1999), 81–89.
  • [4] R. Brawer, Potenzen der Pascalmatrix und eine identit¨at der kombinatorik, Elem. Math., 45 (1990), 107-110.
  • [5] R. Brawer and M. Pirovino, The linear algebra of the Pascal matrix, Linear Algebra Appl., 174 (1992), 13-23.
  • [6] G. S. Call and D. J. Velleman, Pascal’s matrices, Amer. Math. Monthly, 100(4) (1993), 372–376.
  • [7] M. Catalani, Generalized bivariate Fibonacci polynomials, arXiv:math/0211366v2 [math.CO], (2004).
  • [8] M. Catalani, Some formulae for bivariate Fibonacci and Lucas polynomials, arXiv:math/0406323v1 [math.CO], (2004).
  • [9] M. Catalani, Identities for Fibonacci and Lucas polynomials derived from a book of Gould, arXiv:math/0407105v1 [math.CO], (2004).
  • [10] G. S. Cheon, H. Kim and L. W. Shapiro, A generalization of Lucas polynomial sequence, Discrete Appl. Math., 157(5) (2009), 920–927.
  • [11] S. Halıcı and Z. Aky¨uz, On some formulae for bivariate Pell polynomials, Far East Journal of Applied Mathematics, 41(2) (2010), 101-110.
  • [12] E. G. Kocer, Bivariate Vieta–Fibonacci and bivariate Vieta–Lucas polynomials, IOSR Journal of Mathematics, 12(4) 2016, 44-50.
  • [13] G. H. Lawden, Pascal matrices, Math. Gaz., 56(398) (1972), 325-327.
  • [14] G. Y. Lee, J. S. Kim and S. H. Cho, Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math., 130 (2003), 527–534.
  • [15] G. Y. Lee and M. Asci, Some properties of the (p;q)-Fibonacci and (p;q)-Lucas polynomials, J. Appl. Math., 2012 (2012), 264842.
  • [16] A. Nalli and P. Haukkanen, On generalized Fibonacci and Lucas polynomials, Chaos Solitons Fractals, 42(5) (2009), 3179–3186.
  • [17] H. Özimamoğlu and A. Kaya, The linear algebra of the Pell-Lucas matrix, Fundamental Journal of Mathematics and Applications, 7(3), (2024), 158-168
  • [18] B. Sadaoui and A. Krelifa, d-Fibonacci and d-Lucas polynomials, J. Math. Model., 9(3) (2021), 425-436.
  • [19] L. W. Shapiro, S. Getu, W. J. Woan and L .C. Woodson, The Riordan group, Discrete Appl. Math., 34(1-3) (1991), 229–239.
  • [20] S. Uygun, Bivariate Jacobsthal and Jacobsthal Lucas polynomial sequences, J. Math. Computer Sci., 21 (2020), 176-185.
  • [21] Z. Zhang and X.Wang, A factorization of the symmetric Pascal matrix involving the Fibonacci matrix, Discrete Appl. Math., 155(17) (2007), 2371–2376.
  • [22] Z. Zhizheng, The linear algebra of the generalized Pascal matrix, Linear Algebra Appl., 250 (1997), 51–60.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Methods and Special Functions
Journal Section Articles
Authors

Hayrullah Özimamoğlu 0000-0001-7844-1840

Early Pub Date April 29, 2025
Publication Date April 30, 2025
Submission Date December 27, 2023
Acceptance Date December 26, 2024
Published in Issue Year 2025 Volume: 13 Issue: 1

Cite

APA Özimamoğlu, H. (2025). On a New Generalization of Bivariate Fibonacci Polynomials and Their Matrix Representations. Konuralp Journal of Mathematics, 13(1), 78-86.
AMA Özimamoğlu H. On a New Generalization of Bivariate Fibonacci Polynomials and Their Matrix Representations. Konuralp J. Math. April 2025;13(1):78-86.
Chicago Özimamoğlu, Hayrullah. “On a New Generalization of Bivariate Fibonacci Polynomials and Their Matrix Representations”. Konuralp Journal of Mathematics 13, no. 1 (April 2025): 78-86.
EndNote Özimamoğlu H (April 1, 2025) On a New Generalization of Bivariate Fibonacci Polynomials and Their Matrix Representations. Konuralp Journal of Mathematics 13 1 78–86.
IEEE H. Özimamoğlu, “On a New Generalization of Bivariate Fibonacci Polynomials and Their Matrix Representations”, Konuralp J. Math., vol. 13, no. 1, pp. 78–86, 2025.
ISNAD Özimamoğlu, Hayrullah. “On a New Generalization of Bivariate Fibonacci Polynomials and Their Matrix Representations”. Konuralp Journal of Mathematics 13/1 (April 2025), 78-86.
JAMA Özimamoğlu H. On a New Generalization of Bivariate Fibonacci Polynomials and Their Matrix Representations. Konuralp J. Math. 2025;13:78–86.
MLA Özimamoğlu, Hayrullah. “On a New Generalization of Bivariate Fibonacci Polynomials and Their Matrix Representations”. Konuralp Journal of Mathematics, vol. 13, no. 1, 2025, pp. 78-86.
Vancouver Özimamoğlu H. On a New Generalization of Bivariate Fibonacci Polynomials and Their Matrix Representations. Konuralp J. Math. 2025;13(1):78-86.
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