# SOME NEW IDENTITIES CONCERNING THE HORADAM SEQUENCE AND ITS COMPANION SEQUENCE

Keskin, Refik;Siar, Zafer

• Accepted : 2018.12.04
• Published : 2019.01.31
• 152 1

#### Abstract

Let a, b, P, and Q be real numbers with $PQ{\neq}0$ and $(a,b){\neq}(0,0)$. The Horadam sequence $\{W_n\}$ is defined by $W_0=a$, $W_1=b$ and $W_n=PW_{n-1}+QW_{n-2}$ for $n{\geq}2$. Let the sequence $\{X_n\}$ be defined by $X_n=W_{n+1}+QW_{n-1}$. In this study, we obtain some new identities between the Horadam sequence $\{W_n\}$ and the sequence $\{X_n\}$. By the help of these identities, we show that Diophantine equations such as $$x^2-Pxy-y^2={\pm}(b^2-Pab-a^2)(P^2+4),\\x^2-Pxy+y^2=-(b^2-Pab+a^2)(P^2-4),\\x^2-(P^2+4)y^2={\pm}4(b^2-Pab-a^2),$$ and $$x^2-(P^2-4)y^2=4(b^2-Pab+a^2)$$ have infinitely many integer solutions x and y, where a, b, and P are integers. Lastly, we make an application of the sequences $\{W_n\}$ and $\{X_n\}$ to trigonometric functions and get some new angle addition formulas such as $${\sin}\;r{\theta}\;{\sin}(m+n+r){\theta}={\sin}(m+r){\theta}\;{\sin}(n+r){\theta}-{\sin}\;m{\theta}\;{\sin}\;n{\theta},\\{\cos}\;r{\theta}\;{\cos}(m+n+r){\theta}={\cos}(m+r){\theta}\;{\cos}(n+r){\theta}-{\sin}\;m{\theta}\;{\sin}\;n{\theta},$$ and $${\cos}\;r{\theta}\;{\sin}(m+n){\theta}={\cos}(n+r){\theta}\;{\sin}\;m{\theta}+{\cos}(m-r){\theta}\;{\sin}\;n{\theta}$$.

#### References

1. Y. H. Jang and S. P. Jun, Linearlization of generalized Fibonacci sequences, Korean J. Math. 3 (2014), 443-454.
2. D. Kalman and R. Mena, The Fibonacci numbers-exposed, Math. Mag. 76 (2003), no. 3, 167-181.
3. A. F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quart. 3 (1965), 161-176.
4. A. F. Horadam, Tschebyscheff and other functions associated with the sequence {$W_n$(a, b; p, q)}, Fibonacci Quart. 7 (1969), no. 1, 14-22.
5. R. S. Melham, Certain classes of finite sums that involve generalized Fibonacci and Lucas numbers, Fibonacci Quart. 42 (2004), no. 1, 47-54
6. R. S. Melham and A. G. Shannon, Some congruence properties of generalized second-order integer sequences, Fibonacci Quart. 32 (1994), no. 5, 424-428.
7. S. Rabinowitz, Algorithmic manipulation of second-order linear recurrences, Fibonacci Quart. 37 (1999), no. 2, 162-177.
8. A. G. Shannon and A. F. Horadam, Special recurrence relations associated with the {$W_n$(a, b; p, q)}, Fibonacci Quart. 17 (1979), no. 4, 294-299.
9. Z. Siar and R. Keskin, Some new identities concerning generalized Fibonacci and Lucas numbers, Hacet. J. Math. Stat. 42 (2013), no. 3, 211-222.
10. G. Cerda-Morales, On generalized Fibonacci and Lucas numbers by matrix methods, Hacet. J. Math. Stat. 42 (2013), no. 2, 173-179.
11. T.-X. He and P. J.-S. Shiue, On sequences of numbers and polynomials defined by linear recurrence relations of order 2, Int. J. Math. Math. Sci. 2009 (2009), Art. ID 709386, 21 pp.