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Summary: CONTINUOUS SYMMETRIZED SOBOLEV INNER PRODUCTS OF
ORDER N (II)
M. ISABEL BUENO§, FRANCISCO MARCELLÁN¶, AND JORGE SÁNCHEZ-RUIZ
Abstract. 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.
Key words. Sobolev inner product, orthogonal polynomials, semiclassical linear functionals,
recurrence relation, symmetrization process
AMS subject classification. 42C05
1. Introduction. Let us consider the following inner product defined in the lin-
ear space P×P, where P denotes the linear space of polynomials with real coefficients,
p, q s =
R
p q dµ0 +
N
i=1
i
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