  In this paper we study various difference equations related to Jacobi-type
pencils. By a Jacobi-type pencil one means the following pencil: $J_5 - \lambda
J_3$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real
symmetric five-diagonal matrix with positive numbers on the second subdiagonal.
The basic set of solutions for the corresponding $4$-th order difference
equation is constructed. Spectral properties of the truncated pencil and some
special matrix orthogonality relations are investigated. Classical type
orthogonal polynomials satisfying a $4$-th order differential equation are
constructed.
