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Year 2023, Volume: 11 Issue: 1, 24 - 30, 30.04.2023

Abstract

References

  • [1] M. Andeli´c and C. M. da Fonseca, A short proof for a determinantal formula for generalized Fibonacci numbers, Le Matematiche, 74(2), (2019), 363–367.
  • [2] M. Andeli´c and C. M. da Fonseca, On a determinantal formula for derangement numbers, Kragujevac J. Math., 47(6), (2023), 847–850.
  • [3] M. Andeli´c and C. M. da Fonseca, On the constant coefficients of a certain recurrence relation: A simple proof, Heliyon, 7(8), e07764, (2021). https://doi.org/10.1016/j.heliyon.2021.e07764
  • [4] N. Bourbaki, Elements of Mathematics: Functions of a Real Variable: Elementary Theory (Springer, Berlin, 2004). Translated from the 1976 French original by Philip Spain. Elements of Mathematics (Berlin). https://doi.org/10.1007/978-3-642-59315-4
  • [5] N. D. Cahill, J. R. D’Errico, D. A. Narayan and J. Y. Narayan, Fibonacci determinants, College Math. J., 33(3), (2002), 221–225.
  • [6] M. C. Da˘glı and F. Qi, Several closed and determinantal forms for convolved Fibonacci numbers, Discrete Math. Lett., 7, (2021), 14–20.
  • [7] C. M. da Fonseca, On the connection between tridiagonal matrices, Chebyshev polynomails, and Fibonacci numbers, Acta Univ. Sapientiae Math., 12, (2020), 280–286.
  • [8] C. M. da Fonseca, Some comments on the properties of a particular tridiagonal matrix, J. Discret. Math. Sci. Cryptogr., 24(1), (2021), 49–51.
  • [9] C. M. da Fonseca, On a closed form for derangement numbers: an elemantary proof, RACSAM, 114, Article ID:146, (2020). https://doi.org/10.1007/s13398-020-00879-3
  • [10] C. M. da Fonseca, Unifying some Pell and Fibonacci identities, Appl. Math. Comput., 236, (2014), 41–42.
  • [11] B.-N. Guo, E. Polatlı and F. Qi, Determinantal Formulas and Recurrent Relations for Bi-Periodic Fibonacci and Lucas Polynomials. In: Paikray S.K., Dutta H., Mordeson J.N. (eds) New Trends in Applied Analysis and Computational Mathematics. Advances in Intelligent Systems and Computing, vol 1356. Springer, Singapore. https://doi.org/10.1007/978-981-16-1402-6 18
  • [12] A. F. Horadam, Extension of a synthesis for a class of polynomial sequences, Fibonacci Q., 34, (1996), 68–74.
  • [13] C. Kızılates¸, B. C¸ ekim and N.Tu˘glu, T. Kim, New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers, Symmetry, 11, 264, (2019). https://doi.org/10.3390/sym11020264
  • [14] C. Kızılates¸, New families of Horadam numbers associated with finite operators and their applications, Math. Methods Appl. Sci., (2021). https://doi.org/10.1002/mma.7702
  • [15] C. Kızılates¸, Explicit, determinantal, recursive formulas, and generating functions of generalized Humbert–Hermite polynomials via generalized Fibonacci polynomials, Math. Methods Appl. Sci., (2023). https://doi.org/10.1002/mma.9048
  • [16] C. Kızılates¸, W.-S. Du and F. Qi, Several determinantal expressions of generalized Tribonacci polynomials and sequences, Tamkang J. Math., 53, (2022). http://dx.doi.org/10.5556/j.tkjm.53.2022.3743
  • [17] T. Koshy, Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics (Wiley-Interscience, New York), (2001).
  • [18] G. Lee and M. As¸c¸ı, Some Properties of the (p;q)􀀀Fibonacci and (p;q)􀀀Lucas Polynomials, J. Appl. Math., 2012(2012), Article ID: 264842. https://doi.org/10.1155/2012/264842
  • [19] R. S. Martin and J. H. Wilkinson, Handbook series linear algebra: similarity reduction of a general matrix to Hessenberg form, Numer. Math., 12(5), (1968), 349–368. https://doi.org/10.1007/BF02161358
  • [20] E. O¨ zkan and I˙. Altun, Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials, Commun. Algebra, 47(10), (2019), 4020–4030.
  • [21] E. O¨ zkan and M. Tas¸tan, On a new family of Gauss k􀀀Lucas numbers and their polynomials, Asian-Eur. J. Math., 14(6), (2021). https://doi.org/10.1142/S1793557121501011
  • [22] F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput., 268, (2015), 844–858. https://doi.org/10.1016/j.amc.2015.06.123
  • [23] F. Qi, M.C. Da˘glı and W.-S. Du, Determinantal forms and recursive relations of the Delannoy two-functional sequence, Adv. Theory Nonlinear Anal. Appl., 4(3), (2020), 184–193. https://doi.org/10.31197/atnaa.772734
  • [24] J. Ram´ırez, On convolved generalized Fibonacci and Lucas polynomials, Appl. Math. Comput., 229, (2014), 208–213.
  • [25] A. G. Shannon, Fibonacci analogs of the classical polynomials, Math. Mag., 48(3), (1975), 123–130.
  • [26] Y. S¸ims¸ek, Some new families of special polynomials and numbers associated with finite operators, Symmetry, 12(237), (2020), 1–13.
  • [27] Y. S¸ims¸ek, Construction method for generating of special numbers and polynomials arising from analysis of new operators, Math. Meth. Appl. Sci., 41(16), (2018), 6934–6954.
  • [28] N. Terzio˘glu, C. Kızılates¸ and W-S Du, New properties and identities for Fibonacci finite operator quaternions, Mathematics, 10(10), 1719, (2022). https://doi.org/10.3390/math10101719
  • [29] J. Wang, Some new results for the (p;q)􀀀Fibonacci and Lucas polynomials, Adv. Differ. Equ., 2014(2014), Article ID: 64.

On $(p,q)-$Fibonacci Polynomials Connected with Finite Operators

Year 2023, Volume: 11 Issue: 1, 24 - 30, 30.04.2023

Abstract

The aim of this study is to obtain some properties of the $(p,q)-$Fibonacci finite operator polynomials by implementing the finite operator to the $(p,q)-$ Fibonacci polynomials. Firstly, we obtain the Binet formula, generating function, exponential generating function, Poisson generating function, and binomial sum of $(p,q) -$ Fibonacci finite operator polynomials. After that we give determinantal expressions for these finite operator polynomials and their special cases. Lastly, we regain, in a different way, recurrence relation for these finite operator polynomials.

References

  • [1] M. Andeli´c and C. M. da Fonseca, A short proof for a determinantal formula for generalized Fibonacci numbers, Le Matematiche, 74(2), (2019), 363–367.
  • [2] M. Andeli´c and C. M. da Fonseca, On a determinantal formula for derangement numbers, Kragujevac J. Math., 47(6), (2023), 847–850.
  • [3] M. Andeli´c and C. M. da Fonseca, On the constant coefficients of a certain recurrence relation: A simple proof, Heliyon, 7(8), e07764, (2021). https://doi.org/10.1016/j.heliyon.2021.e07764
  • [4] N. Bourbaki, Elements of Mathematics: Functions of a Real Variable: Elementary Theory (Springer, Berlin, 2004). Translated from the 1976 French original by Philip Spain. Elements of Mathematics (Berlin). https://doi.org/10.1007/978-3-642-59315-4
  • [5] N. D. Cahill, J. R. D’Errico, D. A. Narayan and J. Y. Narayan, Fibonacci determinants, College Math. J., 33(3), (2002), 221–225.
  • [6] M. C. Da˘glı and F. Qi, Several closed and determinantal forms for convolved Fibonacci numbers, Discrete Math. Lett., 7, (2021), 14–20.
  • [7] C. M. da Fonseca, On the connection between tridiagonal matrices, Chebyshev polynomails, and Fibonacci numbers, Acta Univ. Sapientiae Math., 12, (2020), 280–286.
  • [8] C. M. da Fonseca, Some comments on the properties of a particular tridiagonal matrix, J. Discret. Math. Sci. Cryptogr., 24(1), (2021), 49–51.
  • [9] C. M. da Fonseca, On a closed form for derangement numbers: an elemantary proof, RACSAM, 114, Article ID:146, (2020). https://doi.org/10.1007/s13398-020-00879-3
  • [10] C. M. da Fonseca, Unifying some Pell and Fibonacci identities, Appl. Math. Comput., 236, (2014), 41–42.
  • [11] B.-N. Guo, E. Polatlı and F. Qi, Determinantal Formulas and Recurrent Relations for Bi-Periodic Fibonacci and Lucas Polynomials. In: Paikray S.K., Dutta H., Mordeson J.N. (eds) New Trends in Applied Analysis and Computational Mathematics. Advances in Intelligent Systems and Computing, vol 1356. Springer, Singapore. https://doi.org/10.1007/978-981-16-1402-6 18
  • [12] A. F. Horadam, Extension of a synthesis for a class of polynomial sequences, Fibonacci Q., 34, (1996), 68–74.
  • [13] C. Kızılates¸, B. C¸ ekim and N.Tu˘glu, T. Kim, New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers, Symmetry, 11, 264, (2019). https://doi.org/10.3390/sym11020264
  • [14] C. Kızılates¸, New families of Horadam numbers associated with finite operators and their applications, Math. Methods Appl. Sci., (2021). https://doi.org/10.1002/mma.7702
  • [15] C. Kızılates¸, Explicit, determinantal, recursive formulas, and generating functions of generalized Humbert–Hermite polynomials via generalized Fibonacci polynomials, Math. Methods Appl. Sci., (2023). https://doi.org/10.1002/mma.9048
  • [16] C. Kızılates¸, W.-S. Du and F. Qi, Several determinantal expressions of generalized Tribonacci polynomials and sequences, Tamkang J. Math., 53, (2022). http://dx.doi.org/10.5556/j.tkjm.53.2022.3743
  • [17] T. Koshy, Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics (Wiley-Interscience, New York), (2001).
  • [18] G. Lee and M. As¸c¸ı, Some Properties of the (p;q)􀀀Fibonacci and (p;q)􀀀Lucas Polynomials, J. Appl. Math., 2012(2012), Article ID: 264842. https://doi.org/10.1155/2012/264842
  • [19] R. S. Martin and J. H. Wilkinson, Handbook series linear algebra: similarity reduction of a general matrix to Hessenberg form, Numer. Math., 12(5), (1968), 349–368. https://doi.org/10.1007/BF02161358
  • [20] E. O¨ zkan and I˙. Altun, Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials, Commun. Algebra, 47(10), (2019), 4020–4030.
  • [21] E. O¨ zkan and M. Tas¸tan, On a new family of Gauss k􀀀Lucas numbers and their polynomials, Asian-Eur. J. Math., 14(6), (2021). https://doi.org/10.1142/S1793557121501011
  • [22] F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput., 268, (2015), 844–858. https://doi.org/10.1016/j.amc.2015.06.123
  • [23] F. Qi, M.C. Da˘glı and W.-S. Du, Determinantal forms and recursive relations of the Delannoy two-functional sequence, Adv. Theory Nonlinear Anal. Appl., 4(3), (2020), 184–193. https://doi.org/10.31197/atnaa.772734
  • [24] J. Ram´ırez, On convolved generalized Fibonacci and Lucas polynomials, Appl. Math. Comput., 229, (2014), 208–213.
  • [25] A. G. Shannon, Fibonacci analogs of the classical polynomials, Math. Mag., 48(3), (1975), 123–130.
  • [26] Y. S¸ims¸ek, Some new families of special polynomials and numbers associated with finite operators, Symmetry, 12(237), (2020), 1–13.
  • [27] Y. S¸ims¸ek, Construction method for generating of special numbers and polynomials arising from analysis of new operators, Math. Meth. Appl. Sci., 41(16), (2018), 6934–6954.
  • [28] N. Terzio˘glu, C. Kızılates¸ and W-S Du, New properties and identities for Fibonacci finite operator quaternions, Mathematics, 10(10), 1719, (2022). https://doi.org/10.3390/math10101719
  • [29] J. Wang, Some new results for the (p;q)􀀀Fibonacci and Lucas polynomials, Adv. Differ. Equ., 2014(2014), Article ID: 64.
There are 29 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Emrah Polatlı 0000-0002-2349-8978

Publication Date April 30, 2023
Submission Date December 2, 2022
Acceptance Date April 7, 2023
Published in Issue Year 2023 Volume: 11 Issue: 1

Cite

APA Polatlı, E. (2023). On $(p,q)-$Fibonacci Polynomials Connected with Finite Operators. Konuralp Journal of Mathematics, 11(1), 24-30.
AMA Polatlı E. On $(p,q)-$Fibonacci Polynomials Connected with Finite Operators. Konuralp J. Math. April 2023;11(1):24-30.
Chicago Polatlı, Emrah. “On $(p,q)-$Fibonacci Polynomials Connected With Finite Operators”. Konuralp Journal of Mathematics 11, no. 1 (April 2023): 24-30.
EndNote Polatlı E (April 1, 2023) On $(p,q)-$Fibonacci Polynomials Connected with Finite Operators. Konuralp Journal of Mathematics 11 1 24–30.
IEEE E. Polatlı, “On $(p,q)-$Fibonacci Polynomials Connected with Finite Operators”, Konuralp J. Math., vol. 11, no. 1, pp. 24–30, 2023.
ISNAD Polatlı, Emrah. “On $(p,q)-$Fibonacci Polynomials Connected With Finite Operators”. Konuralp Journal of Mathematics 11/1 (April 2023), 24-30.
JAMA Polatlı E. On $(p,q)-$Fibonacci Polynomials Connected with Finite Operators. Konuralp J. Math. 2023;11:24–30.
MLA Polatlı, Emrah. “On $(p,q)-$Fibonacci Polynomials Connected With Finite Operators”. Konuralp Journal of Mathematics, vol. 11, no. 1, 2023, pp. 24-30.
Vancouver Polatlı E. On $(p,q)-$Fibonacci Polynomials Connected with Finite Operators. Konuralp J. Math. 2023;11(1):24-30.
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