Discrete-continuous symmetrized Sobolev inner products

Abstract

This paper deals with the bilinear symmetrization problem associated with Sobolev inner products. Let {Q(n)}(n=0)(infinity) be the sequence of monic polynomials orthogonal with respect to a Sobolev inner product of order 1 when one of the measures is discrete and the other one is a nondiscrete positive Borel measure. Furthermore, assume that the supports of such measures are symmetric with respect to the origin so that the corresponding odd moments vanish. We consider the orthogonality properties of the sequences of monic polynomials {P-n}(n=0)(infinity) and {R-n}(n=0)(infinity) such that Q(2n)(x) = P-n(x(2)), Q(2n+1)(x) = xR(n)(x(2)). Moreover, recurrence relations for {P-n}(n=0)(infinity) and {R-n}(n=0)(infinity) are obtained as well as explicit algebraic relations between them.