Of concern is the self-adjoint second-order difference equation \[ -\nabla\left( p_{n}\Delta y_{n}\right) +q_{n}y_{n}=\lambda w_{n}y_{n},\text{ }n\in\left[ 0,N-1\right] ,\;n\in\mathbb{Z}, \] subject to the coupled boundary condition \[ \begin{pmatrix} y_{N-1}\\ \Delta y_{N-1} \end{pmatrix} =e^{i\alpha}K\begin{pmatrix} y_{-1}\\ \Delta y_{-1}\end{pmatrix} \] where \(N\geq2,\) \(\Delta\) is the forward difference operator \(\Delta y_{n}=y_{n+1}-y_{n},\) \(\nabla\) is the backward difference operator \(\nabla y_{n}=y_{n}-y_{n-1},\) \(p_{n},q_{n},w_{n}\) are real numbers with \(p_{-1} =p_{N-1}=1,\) \(-\pi<\alpha\leq\pi\) is a constant parameter, and \(\lambda\) is the spectral parameter. The boundary condition contains as special cases the periodic and antiperiodic boundary conditions.