Continuous symmetrized Sobolev inner products of order N (II)

Given a symmetrized Sobolev inner product of order N, the corresponding sequence of monic orthogonal polynomials {Qn} satisfies Q2n(x)=Pn(x2), Q2n+1(x)=xRn(x2) for certain sequences of monic polynomials {Pn} and {Rn}. In this paper we consider the particular case when all the measures that define the symmetrized Sobolev inner product are equal, absolutely continuous and semiclassical. Under such restrictions, we give explicit algebraic relations between the sequences {Pn} and {Rn}, as well as higher-order recurrence relations that they satisfy.
The work of the authors has been partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grants BFM 2003-06335-C03-02 (M.I.B., F.M., J.S.R.) and BFM2001-3878-C02-01 (J.S.R.), INTAS Research Network NeCCA INTAS 03-51-6637 (F.M.), and the Junta de Andalucía research group FQM-0207 (J.S.R).