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Year 2022, , 145 - 155, 29.12.2022
https://doi.org/10.32323/ujma.1207852

Abstract

References

  • [1] P. Ribenboim. My Numbers, My Friends, Popular Lectures on Number Theory. Springer-Verlag, New York, Inc. 2000.
  • [2] A. Özkoc¸, Some algebraic identities on quadra Fibona-Pell integer sequence, Adv. Differ. Equ., 2015 (2015), Article Number: 148, 10 pages.
  • [3] C. Kızılates¸, On the quadra Lucas-Jacobsthal numbers, Karaelmas Sci. Eng. J., 7(2) (2017), 619-621.
  • [4] O. Dişkaya, H. Menken, On the quadra Fibona-Pell and hexa-Fibona-Pell-Jacobsthal sequences, Mathematical Sciences and Applications E-notes, 7(2) (2019), 149-160.
  • [5] A. Tekcan, A. Özkoc¸, M. Engür, M. E. Özbek, On algebraic identities on a new integer sequence with four parameters, Ars Comb., 127 (2016), 225-238.
  • [6] W. R. Hamilton, Elements of Quaternions, London, England, Green Company, 1866.
  • [7] M. N. S. Swamy, On generalized Fibonacci quaternions, Fibonacci Q., 5 (1973), 547-550.
  • [8] A. F. Horadam, Quaternion recurrence relations, Ulam Q., 2(2) (1993), 22-33.
  • [9] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison Wesley Publishing Company, 1998.
  • [10] K. W. Chen, Identities from the Binomial Transform, J. Number Theory, 124, 142-150, 2007.
  • [11] S. Falcon, A. Plaza, Binomial Transforms of the k􀀀Fibonacci Sequences, Int. J. Nonlinear Sci. Numer. Simul., 10, 1305-1316, 2009.
  • [12] H. Prodinger, Some Information about the Binomial Transform, The Fibonacci Q., 32 (5), 412-415, 1994.
  • [13] H. W. Gould, Series Transformations for Findings Recurrences for Sequences, The Fibonacci Q., 28(2), 166-171, 1990.
  • [14] S. Falcon, Binomial Transform of the Generalized k􀀀Fibonacci Numbers, Commun. Math. Anal., Vol. 10, No. 3, pp. 643–651, 2019
  • [15] C. Kızılates¸, N. Tuğlu, B. Çekim, Binomial transform of quadrapell sequences and quadrapell matrix sequences, J. Sci. Arts, 1(38) (2017), 69-80.
  • [16] T. C¸ etinalp, Kuadra Fibona-Pell kuaterniyon dizileri ¨¨uzerine bazı cebirsel ¨ozdes¸likler, MSc. Thesis, Karamano˘glu Mehmetbey University, 2017.
  • [17] F. Kaplan, A. Özkoc¸ Öztürk, On the binomial transforms of the Horadam quaternion sequences, Math. Meth. Appl. Sci., 45 (2022), 12009-12022.
  • [18] E. Polatlı, On certain properties of quadrapell sequences, Karaelmas Sci. Eng. J., 8(1) (2018), 305-308.

Binomial Transform for Quadra Fibona-Pell Sequence and Quadra Fibona-Pell Quaternion

Year 2022, , 145 - 155, 29.12.2022
https://doi.org/10.32323/ujma.1207852

Abstract

The main object of the study is to consider the binomial transform for quadra Fibona-Pell sequence and quadra Fibona-Pell quaternion. In the paper, which consists of two parts in terms of the results found, the first step was taken for the sequence by defining the binomial transform for the quadra Fibona-Pell sequence in the first part and then finding the recurrence relation of this new binomial transform. Then, the Binet formula, generating function and various sum formulas of the sequence were found. In the second part, the binomial transform is applied for the quadra Fibona-Pell quaternion, which was discussed in a thesis before. Similar results in the first section are covered in the quaternion binomial transform.

References

  • [1] P. Ribenboim. My Numbers, My Friends, Popular Lectures on Number Theory. Springer-Verlag, New York, Inc. 2000.
  • [2] A. Özkoc¸, Some algebraic identities on quadra Fibona-Pell integer sequence, Adv. Differ. Equ., 2015 (2015), Article Number: 148, 10 pages.
  • [3] C. Kızılates¸, On the quadra Lucas-Jacobsthal numbers, Karaelmas Sci. Eng. J., 7(2) (2017), 619-621.
  • [4] O. Dişkaya, H. Menken, On the quadra Fibona-Pell and hexa-Fibona-Pell-Jacobsthal sequences, Mathematical Sciences and Applications E-notes, 7(2) (2019), 149-160.
  • [5] A. Tekcan, A. Özkoc¸, M. Engür, M. E. Özbek, On algebraic identities on a new integer sequence with four parameters, Ars Comb., 127 (2016), 225-238.
  • [6] W. R. Hamilton, Elements of Quaternions, London, England, Green Company, 1866.
  • [7] M. N. S. Swamy, On generalized Fibonacci quaternions, Fibonacci Q., 5 (1973), 547-550.
  • [8] A. F. Horadam, Quaternion recurrence relations, Ulam Q., 2(2) (1993), 22-33.
  • [9] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison Wesley Publishing Company, 1998.
  • [10] K. W. Chen, Identities from the Binomial Transform, J. Number Theory, 124, 142-150, 2007.
  • [11] S. Falcon, A. Plaza, Binomial Transforms of the k􀀀Fibonacci Sequences, Int. J. Nonlinear Sci. Numer. Simul., 10, 1305-1316, 2009.
  • [12] H. Prodinger, Some Information about the Binomial Transform, The Fibonacci Q., 32 (5), 412-415, 1994.
  • [13] H. W. Gould, Series Transformations for Findings Recurrences for Sequences, The Fibonacci Q., 28(2), 166-171, 1990.
  • [14] S. Falcon, Binomial Transform of the Generalized k􀀀Fibonacci Numbers, Commun. Math. Anal., Vol. 10, No. 3, pp. 643–651, 2019
  • [15] C. Kızılates¸, N. Tuğlu, B. Çekim, Binomial transform of quadrapell sequences and quadrapell matrix sequences, J. Sci. Arts, 1(38) (2017), 69-80.
  • [16] T. C¸ etinalp, Kuadra Fibona-Pell kuaterniyon dizileri ¨¨uzerine bazı cebirsel ¨ozdes¸likler, MSc. Thesis, Karamano˘glu Mehmetbey University, 2017.
  • [17] F. Kaplan, A. Özkoc¸ Öztürk, On the binomial transforms of the Horadam quaternion sequences, Math. Meth. Appl. Sci., 45 (2022), 12009-12022.
  • [18] E. Polatlı, On certain properties of quadrapell sequences, Karaelmas Sci. Eng. J., 8(1) (2018), 305-308.
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Arzu Özkoç

Eda Gündüz This is me 0000-0003-1985-6506

Publication Date December 29, 2022
Submission Date November 21, 2022
Acceptance Date December 26, 2022
Published in Issue Year 2022

Cite

APA Özkoç, A., & Gündüz, E. (2022). Binomial Transform for Quadra Fibona-Pell Sequence and Quadra Fibona-Pell Quaternion. Universal Journal of Mathematics and Applications, 5(4), 145-155. https://doi.org/10.32323/ujma.1207852
AMA Özkoç A, Gündüz E. Binomial Transform for Quadra Fibona-Pell Sequence and Quadra Fibona-Pell Quaternion. Univ. J. Math. Appl. December 2022;5(4):145-155. doi:10.32323/ujma.1207852
Chicago Özkoç, Arzu, and Eda Gündüz. “Binomial Transform for Quadra Fibona-Pell Sequence and Quadra Fibona-Pell Quaternion”. Universal Journal of Mathematics and Applications 5, no. 4 (December 2022): 145-55. https://doi.org/10.32323/ujma.1207852.
EndNote Özkoç A, Gündüz E (December 1, 2022) Binomial Transform for Quadra Fibona-Pell Sequence and Quadra Fibona-Pell Quaternion. Universal Journal of Mathematics and Applications 5 4 145–155.
IEEE A. Özkoç and E. Gündüz, “Binomial Transform for Quadra Fibona-Pell Sequence and Quadra Fibona-Pell Quaternion”, Univ. J. Math. Appl., vol. 5, no. 4, pp. 145–155, 2022, doi: 10.32323/ujma.1207852.
ISNAD Özkoç, Arzu - Gündüz, Eda. “Binomial Transform for Quadra Fibona-Pell Sequence and Quadra Fibona-Pell Quaternion”. Universal Journal of Mathematics and Applications 5/4 (December 2022), 145-155. https://doi.org/10.32323/ujma.1207852.
JAMA Özkoç A, Gündüz E. Binomial Transform for Quadra Fibona-Pell Sequence and Quadra Fibona-Pell Quaternion. Univ. J. Math. Appl. 2022;5:145–155.
MLA Özkoç, Arzu and Eda Gündüz. “Binomial Transform for Quadra Fibona-Pell Sequence and Quadra Fibona-Pell Quaternion”. Universal Journal of Mathematics and Applications, vol. 5, no. 4, 2022, pp. 145-5, doi:10.32323/ujma.1207852.
Vancouver Özkoç A, Gündüz E. Binomial Transform for Quadra Fibona-Pell Sequence and Quadra Fibona-Pell Quaternion. Univ. J. Math. Appl. 2022;5(4):145-5.

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