The holonomic equation of the Laguerre–Sobolev-type orthogonal polynomials: a non-diagonal case

In this paper, we consider the Sobolev-type inner product where p and q are polynomials with real coefficients, and A is a positive semi-definite matrix. First, we consider a multiplication operator that is symmetric with respect to the above inner product. As a consequence, we prove that the sequence of monic polynomials orthogonal with respect to the above inner product satisfies a five-term recurrence relation. On the other hand, we obtain raising and lowering operators associated with them. As a consequence, a holonomic equation satisfied by these polynomials is given.