Difference equations related to Jacobi-type pencils

Abstract

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.

26 pages. arXiv admin note: text overlap with arXiv:1706.02391