Some new identities concerning the Horadam sequence and its companion sequence

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 $\left\{ W_{n}\right\} $ and the sequence $\left\{ X_{n}\right\} $. By the help of these identities, we show that Diophantine equations such as \begin{align*} 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}), \end{align*} 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 $\left\{ W_{n}\right\} $ and $\left\{ X_{n}\right\} $ 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 \[ \cos r\theta \sin (m+n)\theta =\cos (n+r)\theta \sin m\theta +\cos (m-r)\theta \sin n\theta . \] KCI Citation Count: 1