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

In this paper, we consider the Sobolev-type inner product < p, q >(S) = integral(infinity)(0) p(x)q(x)x(alpha)e(-x) dx + P(0)(t)AQ(0), alpha > -1 where p and q are polynomials with real coefficients, A = (M lambda 0 M1 lambda), P(0) = (p(0) p'(0)), Q(0) = (q(0) q'(0)), 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.