An analysis of the recurrence coefficients for symmetric sobolev-type orthogonal polynomials

Abstract

In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:〈p, q〉s =∫Rp(x)q(x)dµ(x) + M0 p(0)q(0) + M1 p′ (0)q′ (0), where p, q are polynomials, M0, M1 are non-negative real numbers and µ is a symmetric positive measure. These include a five-term recurrence relation, a three-term recurrence relation with rational coefficients, and an explicit expression for its norms. Moreover, we use these results to deduce asymptotic properties for the recurrence coefficients and a nonlinear difference equation that they satisfy, in the particular case when dµ(x) = e−x4 dx.