Öz
In this study, we obtain the relationship between two different generalized Fibonacci polynomials ($F_{k,n}(t)$ and $F_{k,n}(s)$). We discuss some of the special cases of $F_{k,n}(t)$ and $F_{k,n}(s)$, and we show that the obtained results are valid in these special cases. Since $F_{k,n}(s)$ is a new polynomial obtained by a different selection of the coefficients of the core polynomial used to define $F_{k,n}(t)$, our results will provide a new perspective on this issue. This perspective allows us to generalize classical results, such as the relationship between number sequences, the connection between this relationship and the coefficients of the core polynomial, and the method of obtaining these sequences using matrices.
Anahtar Kelimeler
Generalized Fibonacci polynomials Pell sequence Tribonacci sequence Tetrabacci Sequence
Kaynakça
- Cahill, N.D., D’Errico, J.R., Narayan, D.A., Narayan, J.Y., Fibonacci determinants, College Math. J., 33(2002), 221–225.
- Chung, C-L., Some Polynomial Sequence Relations, Mathematics, 7(8)(2019), 750.
- Frontczak R., Relations for Generalized Fibonacci and tribonacci sequences, Notes on Number Theory and Discrete Mathematics, 25(1)(2019), 178–192.
- Frontczak R., Goy T., Shattuck M., Fibonacci-Lucas-Pell-Jacobsthal relations, Annales Mathematicae et Informaticae, 55(2022), 28–48.
- Goy, T., Shattuck, M., Determinants of Toeplitz-Hessenberg matrices with generalized Fibonacci entries, Notes on Number Theory and Discrete Mathematics, 25(2019), 83–95.
- Kaygısız, K., Şahin, A., A new method to compute the terms of generalized order-k Fibonacci numbers, Journal of Number Theory, 133(9)(2013), 3119–3126.
- Kaygisiz, K., Sahin, A., Determinantal and permanental representations of Fibonacci type numbers and polynomials, Rocky Mountain Journal of Mathematics, 46(1)(2016), 227–242.
- Kuhapatanakul, K., Thongsing, K., Generalizations of the Fibonacci–Lucas relations, The American Mathematical Monthly, 126(1)(2019), 81.
- Kılıc, E., Tasci, D., Haukkanen, P., On the generalized Lucas sequences by Hessenberg matrices, Ars Combin., 95(2010), 383–395.
- Leerawat, U., Daowsud, K., Determinants of some Hessenberg matrices with generating functions, Special Matrices, 11(2023), 1–8.
- Li, H-C., On Fibonacci-Hessenberg matrices and the Pell and Perrin numbers, Appl. Math. Comput., 218(17)(2012), 8353–8358.
- Li, H., MacHenry, T., Permanents and determinants, weighted isobaric polynomials, and integer sequences, Journal of Integer Sequences, 16(2013).
- MacHenry, T., A subgroup of the group of units in the ring of arithmetic functions, Rocky Mountain J. Math., 29(3)(1999), 1055–1065.
- MacHenry, T., Generalized Fibonacci and Lucas polynomials and multiplicative arithmetic functions, Fibonacci Quart., 38(2000), 17–24.
- MacHenry, T., Wong, K., A Correspondence between isobaric rings and multiplicative arithmetic funtions, Rocky Mountain J. Math., 42(4)(2012), 1247–1290.
- OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences, Retrieved Dec. 19, 2022, from http://oeis.org
- Şahin, A., Ram´ırez, J.L., Determinantal and permanental representations of convolved Lucas polynomials, Applied Mathematics and Computation, 281(2016), 314–322.
- Şahin, A., Relations between Derangement and Factorial Numbers, Integers, 22(2022).
- Zhong, J, Yao, J, Chung, C-L., A note on incomplete Fibonacci–Lucas relations, Symmetry, 15(12)(2023), 2113.
Öz
Kaynakça
- Cahill, N.D., D’Errico, J.R., Narayan, D.A., Narayan, J.Y., Fibonacci determinants, College Math. J., 33(2002), 221–225.
- Chung, C-L., Some Polynomial Sequence Relations, Mathematics, 7(8)(2019), 750.
- Frontczak R., Relations for Generalized Fibonacci and tribonacci sequences, Notes on Number Theory and Discrete Mathematics, 25(1)(2019), 178–192.
- Frontczak R., Goy T., Shattuck M., Fibonacci-Lucas-Pell-Jacobsthal relations, Annales Mathematicae et Informaticae, 55(2022), 28–48.
- Goy, T., Shattuck, M., Determinants of Toeplitz-Hessenberg matrices with generalized Fibonacci entries, Notes on Number Theory and Discrete Mathematics, 25(2019), 83–95.
- Kaygısız, K., Şahin, A., A new method to compute the terms of generalized order-k Fibonacci numbers, Journal of Number Theory, 133(9)(2013), 3119–3126.
- Kaygisiz, K., Sahin, A., Determinantal and permanental representations of Fibonacci type numbers and polynomials, Rocky Mountain Journal of Mathematics, 46(1)(2016), 227–242.
- Kuhapatanakul, K., Thongsing, K., Generalizations of the Fibonacci–Lucas relations, The American Mathematical Monthly, 126(1)(2019), 81.
- Kılıc, E., Tasci, D., Haukkanen, P., On the generalized Lucas sequences by Hessenberg matrices, Ars Combin., 95(2010), 383–395.
- Leerawat, U., Daowsud, K., Determinants of some Hessenberg matrices with generating functions, Special Matrices, 11(2023), 1–8.
- Li, H-C., On Fibonacci-Hessenberg matrices and the Pell and Perrin numbers, Appl. Math. Comput., 218(17)(2012), 8353–8358.
- Li, H., MacHenry, T., Permanents and determinants, weighted isobaric polynomials, and integer sequences, Journal of Integer Sequences, 16(2013).
- MacHenry, T., A subgroup of the group of units in the ring of arithmetic functions, Rocky Mountain J. Math., 29(3)(1999), 1055–1065.
- MacHenry, T., Generalized Fibonacci and Lucas polynomials and multiplicative arithmetic functions, Fibonacci Quart., 38(2000), 17–24.
- MacHenry, T., Wong, K., A Correspondence between isobaric rings and multiplicative arithmetic funtions, Rocky Mountain J. Math., 42(4)(2012), 1247–1290.
- OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences, Retrieved Dec. 19, 2022, from http://oeis.org
- Şahin, A., Ram´ırez, J.L., Determinantal and permanental representations of convolved Lucas polynomials, Applied Mathematics and Computation, 281(2016), 314–322.
- Şahin, A., Relations between Derangement and Factorial Numbers, Integers, 22(2022).
- Zhong, J, Yao, J, Chung, C-L., A note on incomplete Fibonacci–Lucas relations, Symmetry, 15(12)(2023), 2113.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Cebir ve Sayı Teorisi |
| Bölüm | Makaleler |
| Yazarlar | |
| Yayımlanma Tarihi | 31 Aralık 2024 |
| Yayımlandığı Sayı | Yıl 2024 Cilt: 16 Sayı: 2 |