On the orthogonality of solutions for higher-order non-Hermitian difference equations

In this paper we study higher-order difference equations which can be written as follows: $$\mathbf{J}(y_0,y_1,\mathrm{.}\mathrm{.}\mathrm{.}{)}^T={\lambda }^N(y_0,y_1,\mathrm{.}\mathrm{.}\mathrm{.}{)}^T,$$ where $\mathbf{J}$ is a $(2N+1)$-diagonal bounded banded matrix ($\mathbf{J}=(g_{m,n}{)}_{m,n=0}^{\mathrm{\infty }}$, $\mathrm{\mid }{g}_{m,n}\mathrm{\mid }<C$, $C>0$; and $g_{k,l}=0$ if $\mathrm{\mid }k-l\mathrm{\mid }>N$), $y_j$s are unknowns, $\lambda$ is a complex parameter, $N\in \mathbb{N}$. It is assumed that all $g_{k,k+N}$ and $g_{l-N,l}$ are nonzero. Two special cases are considered: \noindent $\mathit{\text{Case}}\mathit{\text{A}}$: The matrix $\mathbf{J}$ is complex symmetric, i.e. $\mathbf{J}={\mathbf{J}}^T$. \noindent $\mathit{\text{Case}}\mathit{\text{B}}$: The matrix $\mathbf{J}$ is such that $g_{k,k+N}=1$, $k=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}$. Notice that this condition can be attained by changing $y_j$s by their multiples. In both cases there exists a $\mathit{\text{positive}}$ matrix measure $M$ on a circle in the complex plane such that polynomial solutions satisfy some orthogonality relations. Namely, in case~A this is related to a $J$-orthogonality in the Hilbert space $L^2(M)$ ($J$ is a complex conjugation). In case~B we have a left $J$-orthogonality in $L^2(M)$. As a tool, a related matrix moment problem is studied. A complex rank-one perturbation of a free Jacobi matrix is discussed.