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

Abstract

In this paper we study higher-order difference equations which can be written as follows: $$ \mathbf{J} (y_0,y_1,...)^T = \lambda^N (y_0,y_1,...)^T, $$ where $\mathbf{J}$ is a $(2N+1)$-diagonal bounded banded matrix ($\mathbf{J}=(g_{m,n})_{m,n=0}^\infty$, $| g_{m,n} |< C$, $C>0$; and $g_{k,l}=0$ if $|k-l|>N$), $y_j$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 \textit{Case A}: The matrix $\mathbf{J}$ is complex symmetric, i.e. $\mathbf{J} = \mathbf{J}^T$. \noindent \textit{Case B}: The matrix $\mathbf{J}$ is such that $g_{k,k+N}=1$, $k=0,1,2,...$. Notice that this condition can be attained by changing $y_j$s by their multiples. In both cases there exists a \textit{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.