An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials

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) =integral(R)p(x)q(x)d mu(x) + M(0)p(0)q(0) + M1p '(0)q '(0), where p,q are polynomials, M-0, M-1 are non-negative real numbers and mu 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 d mu(x) = e(-x4)dx.

Automated summary

In this contribution, algebraic properties associated with polynomials orthogonal with respect to a Sobolev-type inner product are obtained. These properties include recurrence relations, explicit expressions for norms, and asymptotic properties for the recurrence coefficients and a nonlinear difference equation. The results are deduced specifically for the case when the measure is e^(-x^4)dx.