On the explicit Binet formula of the generalized ${{2}^{nd}}$ orders Recursive relation

Abstract

In this paper, second-order generalized linear recurrence relations of the form ${{V}_{n}}\left( {p}_{1},{p}_{2}, {V}_{1},{V}_{2}\right)={{p}_{1}}{{V}_{n-1}}+{p}_{2}{{V}_{n-2}}$ , where ${{p}_{1}},{{p}_{2}},$ ${{V}_{1}}\left( =a \right)$ and $ {{V}_{2}}\left( =b \right) $ are arbitrary integers, are studied to derive Binet-like formulas in simplified and comprehensive generalized forms. By imposing specific constraints on the coefficients $\left( {{p}_{1}},{{p}_{2}} \right)$ and the initial terms $\left( {{V}_{1}},{{V}_{2}} \right)$, various well-known existing formulas, such as those for classical Fibonacci and Lucas sequences, emerge as special cases of this generalization.

Keywords

• ${{2}^{nd}}$ order Recursive relations
• generaized generating function
• Explicit Binet formulas

Burada genelleştirilmiş ${{2}^{nd}}$ emir özyinelemeli ilişkisinin açık Binet formülü verilmiştir.

Abstract

Bu yazıda, ${{V__{n}}\left( {p__{1},{p__{2}, {V__{1} formundaki ikinci dereceden genelleştirilmiş doğrusal yineleme ilişkileri ,{V__{2}\right)={{p__{1}}{{V__{n-1}}+{p__{2}{{V__{n-2} }$ , burada ${{p__{1}},{{p__{2}},$ ${{V__{1}}\left( =a \right)$ ve $ {{V} _{2}}\left( =b \right) $ rastgele tamsayılardır ve basitleştirilmiş ve kapsamlı genelleştirilmiş formlarda Binet benzeri formüller türetmek için incelenmiştir. $\left( {{p__{1}},{{p__{2}} \right)$ katsayılarına ve $\left( {{V__{1}) başlangıç ​​terimlerine belirli kısıtlamalar uygulayarak },{{V__{2}} \right)$, klasik Fibonacci ve Lucas dizileri için olanlar gibi iyi bilinen çeşitli mevcut formüller, bu genellemenin özel durumları olarak ortaya çıkar.

Keywords

• ${{2}^{nd}}$ Sıra Özyinelemeli İlişkiler
• Genelleştirilmiş Üreten Fonksiyonlar
• Açık Binet Formülleri

References

1. Atkins, J., & Geist, R. (1987). Fibonacci numbers and computer algorithms. The College Mathematics Journal, 18, 328–337.
2. Basin, S., & Hoggatt, V. (1963). A primer on the Fibonacci sequence Part I. The Fibonacci Quarterly, 1, 61–68.
3. Bilgici, G. (2014). New generalizations of Fibonacci and Lucas sequences. American Mathematical Science, 8 (29), 429–1437.
4. Dresden, G. P. B., & Du, Z. (2014). A simplified Binet formula for k-generalized Fibonacci numbers. Journal of Integer Sequences, 17 (4), Article 14.4.7.
5. Edson, M., & Yayenie, O. (2009). A new generalization of Fibonacci sequences and extended Binet’s. Integers, 9, 639–654.
6. Ferguson, D. E. (1966). An expression for generalized Fibonacci numbers. The Fibonacci Quarterly, 4, 270–273.
7. George, A. H. (1969). Some formulae for the Fibonacci sequence with generalizations. The Fibonacci Quarterly, 113–130.
8. Horadam, A. F. (1961). A generalized Fibonacci sequence. The American Mathematical Monthly, 68, 455–459.

9. Jaiswal, D. V. (1969). On a generalized Fibonacci sequence. Labdev Journal of Science and Technology Part A, 7, 67–71.
10. Klein, S. T. (1991). Combinatorial representation of generalized Fibonacci numbers. The Fibonacci Quarterly, 29, 124–131.
11. Koshy, T. (2001). Fibonacci and Lucas numbers with applications. Wiley.
12. Krassimir, A. T., Liliya, A. C., & Dimitar, S. D. (1985). A new perspective to the generalization of the Fibonacci sequence. The Fibonacci Quarterly, 23 (1), 21–28.
13. Lee, G. Y., Lee, S. G., & Shin, H. G. (1997). On the k-generalized Fibonacci matrix Qk. Linear Algebra and Its Applications, 251, 73–88.
14. Lee, G. Y., Lee, S. G., Kim, J. S., & Shin, H. K. (2001). The Binet formula and representations of k-generalized Fibonacci numbers. The Fibonacci Quarterly, 39 (2), 158–164.
15. Miller, M. D. (1971). On generalized Fibonacci numbers. The American Mathematical Monthly, 78, 1108–1109.
16. Pond, J. C. (1968). Generalized Fibonacci summations. The Fibonacci Quarterly, 6, 97–108.
17. Sburlati, G. (2002). Generalized Fibonacci sequences and linear congruence. The Fibonacci Quarterly, 40, 446–452.
18. Spickerman, W. R., & Joyner, R. N. (1984). Binet’s formula for the recursive sequence of order k. The Fibonacci Quarterly, 22, 327–331.
19. Verma, K. L. (in press). On the matrix representation of generalized sequence and applications. J. Mathematical Sciences. Comp. Math., 6 (1).
20. Waddill, M., & Sacks, L. (1967). Another generalized Fibonacci sequence. The Fibonacci Quarterly, 5 (3), 209–222.
21. Wolfram, A. (1998). Solving generalized Fibonacci recurrences. The Fibonacci Quarterly, 36, 129–145. 140