María Álvarez de Morales

Sobolev orthogonality for the Gegenbauer polynomials { C n (-N+1/2) } n ≥0

Abstract In this work, we obtain the property of Sobolev orthogonality for the Gegenbauer polynomials {C n (−N+ 1 2 ) } n ⩾0 , with N ⩾1 a given nonnegative integer, that is, we show that they are orthogonal with respect to some inner product involving derivatives. The Sobolev orthogonality can be used to recover properties of these Gegenbauer polynomials. For instance, we can obtain linear relations with the classical Gegenbauer polynomials.

Orthogonality of the Jacobi polynomials with negative integer parameters

It is well known that the Jacobi polynomials Pn(α,β)(x) are orthogonal with respect to a quasi-definite linear functional whenever α, β, and α + β + 1 are not negative integer numbers. Recently, Sobolev orthogonality for these polynomials has been obtained for α a negative integer and β not a negative integer and also for the case α = β negative integer numbers.In this paper, we give a Sobolev orthogonality for the Jacobi polynomials in the remainder cases.

A matrix Rodrigues formula for classical orthogonal polynomials in two variables

Classical orthogonal polynomials in one variable can be characterized as the only orthogonal polynomials satisfying a Rodrigues formula. In this paper, using the second kind Kronecker power of a matrix, a Rodrigues formula is introduced for classical orthogonal polynomials in two variables.

Bivariate orthogonal polynomials in the Lyskova class

Classical orthogonal polynomials in two variables can be characterized as the polynomial solutions of a matrix second-order partial differential equation involving matrix polynomial coefficients. In this work, we study classical orthogonal polynomials in two variables whose partial derivatives satisfy again a second-order partial differential equation of the same type.

Orthogonal Polynomials Associated with a Δ-Sobolev Inner Product

We will study the family of polynomials which are orthogonal with respect to the discrete inner product involving difference operators

NON-STANDARD ORTHOGONALITY FOR MEIXNER POLYNOMIALS

(\,f,g)^{(K)}_\Delta=\sum^K_{m,k=0} \langle \lambda_{m,k}u,\Delta{^m}f\Delta^{k}g\rangle, \quad K \ge 0

Nondiagonal Hermite–Sobolev Orthogonal Polynomials

In this paper, we consider bivariate orthogonal polynomials associated with a quasi-definite moment functional which satisfies a Pearson-type partial differential equation. For these polynomials differential properties are obtained. In particular, we deduce some structure and orthogonality relations for the successive partial derivatives of the polynomials.