In this study, we investigate a generalization of the modified Pell sequence, which is called $(s,t)$-modified Pell sequence. By considering this sequence, we define the matrix sequence whose elements are $(s,t)$-modified Pell numbers. Furthermore, we define various binomial transforms for modified $ (s,t)$-Pell matrix sequence. Finally, we give some relationships for $(s,t)$-modified Pell matrix sequences such as Binet formulas, the generating functions, and some sum formulas.
[1] S. Falcon, A. Plaza, Binomial transforms of the k-Fibonacci sequence, Int. J. Nonlinear Sci. Numer. Simul., 10(11-12) (2009), 1527-1538.
[2] N. Yilmaz, N. Taskara, Binomial sransforms of the Padovan and Perrin matrix sequences, Abstr. Appl. Anal., 2013 (2013), Article ID 497418, 7 pages.
[3] P. Bhadouria, D. Jhala, B. Singh, Binomial transforms of the k-Lucas sequence, J. Math. Comput. Sci., 8(1) (2014), 81-92.
[4] S. Uygun, A. Erdo˘gdu, Binominal transforms of k-Jacobsthal sequences, J. Math. Comput. Sci., 7(6) (2017), 1100-1114.
[5] C. Kızılateş, N. Tuglu, B. Çekim, Binomial transform of quadrapell sequences and quadrapell matrix sequences, J. Sci. Arts, 1(38) (2017), 69-80.
[6] S. Uygun, The binomial transforms of the generalized (s; t) -Jacobsthal matrix sequence, Int. J. Adv. Appl. Math. Mech., 6(3) (2019), 14-20.
[7] Y. Kwon, Binomial transforms of the modified k-Fibonacci-like sequence, Int. J. Math. Comput. Sci., 14(1) (2019), 47-59.
[8] S. Uygun, Binominal transforms of k-Jacobsthal Lucas sequences, Rom. J. Math. Comput. Sci., 2(10) (2020), 43-54.
[9] N. Yılmaz, Binomial transforms of the Balancing and Lucas-Balancingpolynomials, Contributions Discrete Math., 15(3) (2020), 133-144.
[10] Y. Soykan, Binomial transform of the generalized tribonacci sequence, Asian Res. J. Math., 16(10) (2020), 26-55.
[11] Y. Soykan, Binomial transform of the generalized third orde Pell sequence, Commun. Math. Appl., 12(1) (2021), 71-94.
[12] Y. Soykan, Binomial transform of the generalized fourth order Pell sequence, Arch. Current Res. Internat., 21(6) (2021),9-31.
[13] Y. Soykan, On binomial transform of the generalized fifth order Pell Sequence, Asian J. Adv. Res. Rep., 5(9) (2021), 18-29.
[14] Y. Soykan, Notes on binomial transform of the generalized Narayana sequence, Earthline J. Math. Sci., 7(1) (2021), 77-111.
[15] Y. Soykan, Binomial transform of the generalized pentanacci sequence, Asian Res. J. Current Sci., 3(1) (2021), 209-231.
[16] Y. Soykan, E. Taşdemir, N. Ozmen, On binomial transform of the generalized Jacobsthal-Padovan numbers, Int. J. Nonlinear Anal. Appl., 14(1) (2023), 643-666.
[17] K. N. Boyadzhiev, Notes on the Binomial Transform: Theory and Table with Appendix on Stirling Transform, World Scientific Publishing, Singapore, 2018.
[18] S. K. Ghosal, J.K. Mandal, Binomial transform based fragile watermarking for image authentication, J. Inf. Secur. Appl., 19(4-5) (2014), 272-281.
[19] A. A. Wani, P. Catarino, S. Halici, On a study of generalized Pell sequence and its matrix sequence, Punjab Univ. J. Math., 51(9) (2020), 17-39.
[20] A. Özkoç Öztuürk, E. Gündüz, Binomial transform for quadra Fibona-Pell sequence and Quadra Fibona-Pell quaternion, Univ. J. Math. Appl., 5(4) (2022), 145-155.
[21] A. F. Horadam, Pell identities, Fibonacci Quart., 9(3) (1971), 245-252.
[22] N. Karaaslan, T. Yağmur, (s; t)-Modified Pell sequence and its matrix representation, Erzincan Univ. J. Sci. Tech., 12(2) (2019), 863-873.
The Properties of Binomial Transforms for Modified $(s,t)$-Pell Matrix Sequence
Year 2024,
Volume: 7 Issue: 3, 168 - 177, 29.09.2024
In this study, we investigate a generalization of the modified Pell sequence, which is called $(s,t)$-modified Pell sequence. By considering this sequence, we define the matrix sequence whose elements are $(s,t)$-modified Pell numbers. Furthermore, we define various binomial transforms for modified $ (s,t)$-Pell matrix sequence. Finally, we give some relationships for $(s,t)$-modified Pell matrix sequences such as Binet formulas, the generating functions, and some sum formulas.
[1] S. Falcon, A. Plaza, Binomial transforms of the k-Fibonacci sequence, Int. J. Nonlinear Sci. Numer. Simul., 10(11-12) (2009), 1527-1538.
[2] N. Yilmaz, N. Taskara, Binomial sransforms of the Padovan and Perrin matrix sequences, Abstr. Appl. Anal., 2013 (2013), Article ID 497418, 7 pages.
[3] P. Bhadouria, D. Jhala, B. Singh, Binomial transforms of the k-Lucas sequence, J. Math. Comput. Sci., 8(1) (2014), 81-92.
[4] S. Uygun, A. Erdo˘gdu, Binominal transforms of k-Jacobsthal sequences, J. Math. Comput. Sci., 7(6) (2017), 1100-1114.
[5] C. Kızılateş, N. Tuglu, B. Çekim, Binomial transform of quadrapell sequences and quadrapell matrix sequences, J. Sci. Arts, 1(38) (2017), 69-80.
[6] S. Uygun, The binomial transforms of the generalized (s; t) -Jacobsthal matrix sequence, Int. J. Adv. Appl. Math. Mech., 6(3) (2019), 14-20.
[7] Y. Kwon, Binomial transforms of the modified k-Fibonacci-like sequence, Int. J. Math. Comput. Sci., 14(1) (2019), 47-59.
[8] S. Uygun, Binominal transforms of k-Jacobsthal Lucas sequences, Rom. J. Math. Comput. Sci., 2(10) (2020), 43-54.
[9] N. Yılmaz, Binomial transforms of the Balancing and Lucas-Balancingpolynomials, Contributions Discrete Math., 15(3) (2020), 133-144.
[10] Y. Soykan, Binomial transform of the generalized tribonacci sequence, Asian Res. J. Math., 16(10) (2020), 26-55.
[11] Y. Soykan, Binomial transform of the generalized third orde Pell sequence, Commun. Math. Appl., 12(1) (2021), 71-94.
[12] Y. Soykan, Binomial transform of the generalized fourth order Pell sequence, Arch. Current Res. Internat., 21(6) (2021),9-31.
[13] Y. Soykan, On binomial transform of the generalized fifth order Pell Sequence, Asian J. Adv. Res. Rep., 5(9) (2021), 18-29.
[14] Y. Soykan, Notes on binomial transform of the generalized Narayana sequence, Earthline J. Math. Sci., 7(1) (2021), 77-111.
[15] Y. Soykan, Binomial transform of the generalized pentanacci sequence, Asian Res. J. Current Sci., 3(1) (2021), 209-231.
[16] Y. Soykan, E. Taşdemir, N. Ozmen, On binomial transform of the generalized Jacobsthal-Padovan numbers, Int. J. Nonlinear Anal. Appl., 14(1) (2023), 643-666.
[17] K. N. Boyadzhiev, Notes on the Binomial Transform: Theory and Table with Appendix on Stirling Transform, World Scientific Publishing, Singapore, 2018.
[18] S. K. Ghosal, J.K. Mandal, Binomial transform based fragile watermarking for image authentication, J. Inf. Secur. Appl., 19(4-5) (2014), 272-281.
[19] A. A. Wani, P. Catarino, S. Halici, On a study of generalized Pell sequence and its matrix sequence, Punjab Univ. J. Math., 51(9) (2020), 17-39.
[20] A. Özkoç Öztuürk, E. Gündüz, Binomial transform for quadra Fibona-Pell sequence and Quadra Fibona-Pell quaternion, Univ. J. Math. Appl., 5(4) (2022), 145-155.
[21] A. F. Horadam, Pell identities, Fibonacci Quart., 9(3) (1971), 245-252.
[22] N. Karaaslan, T. Yağmur, (s; t)-Modified Pell sequence and its matrix representation, Erzincan Univ. J. Sci. Tech., 12(2) (2019), 863-873.
Uygun, Ş., & Haklıdır, O. (2024). The Properties of Binomial Transforms for Modified (s,t)-Pell Matrix Sequence. Communications in Advanced Mathematical Sciences, 7(3), 168-177. https://doi.org/10.33434/cams.1524027
AMA
Uygun Ş, Haklıdır O. The Properties of Binomial Transforms for Modified (s,t)-Pell Matrix Sequence. Communications in Advanced Mathematical Sciences. September 2024;7(3):168-177. doi:10.33434/cams.1524027
Chicago
Uygun, Şükran, and Ozan Haklıdır. “The Properties of Binomial Transforms for Modified (s,t)-Pell Matrix Sequence”. Communications in Advanced Mathematical Sciences 7, no. 3 (September 2024): 168-77. https://doi.org/10.33434/cams.1524027.
EndNote
Uygun Ş, Haklıdır O (September 1, 2024) The Properties of Binomial Transforms for Modified (s,t)-Pell Matrix Sequence. Communications in Advanced Mathematical Sciences 7 3 168–177.
IEEE
Ş. Uygun and O. Haklıdır, “The Properties of Binomial Transforms for Modified (s,t)-Pell Matrix Sequence”, Communications in Advanced Mathematical Sciences, vol. 7, no. 3, pp. 168–177, 2024, doi: 10.33434/cams.1524027.
ISNAD
Uygun, Şükran - Haklıdır, Ozan. “The Properties of Binomial Transforms for Modified (s,t)-Pell Matrix Sequence”. Communications in Advanced Mathematical Sciences 7/3 (September 2024), 168-177. https://doi.org/10.33434/cams.1524027.
JAMA
Uygun Ş, Haklıdır O. The Properties of Binomial Transforms for Modified (s,t)-Pell Matrix Sequence. Communications in Advanced Mathematical Sciences. 2024;7:168–177.
MLA
Uygun, Şükran and Ozan Haklıdır. “The Properties of Binomial Transforms for Modified (s,t)-Pell Matrix Sequence”. Communications in Advanced Mathematical Sciences, vol. 7, no. 3, 2024, pp. 168-77, doi:10.33434/cams.1524027.
Vancouver
Uygun Ş, Haklıdır O. The Properties of Binomial Transforms for Modified (s,t)-Pell Matrix Sequence. Communications in Advanced Mathematical Sciences. 2024;7(3):168-77.