On the solution of a second order linear homogeneous difference equation with variable coefficients (Q1378584)

The author presents a complete closed form solution of the second-order linear homogeneous difference equation \(y_{n+2}= -A_ny_{n+1}- B_ny_n\), \(n\geq 1\), with integral index \(n\), variable complex coefficients \(A_n\) and \(B_n\neq 0\), and complex initial values \(y_1\), \(y_2\). Both the linearly independent solutions are expressed solely in terms of the coefficients. Expressions for \(\cos n\theta\) and \(\sin(n+1)\theta/\sin\theta\) as polynomials of degree \([n/2]\) in \(\cos\theta\) are also obtained from the solution.