On the Christoffel–Darboux formula for generalized matrix orthogonal polynomials (original) (raw)

Another Christoffel--Darboux Formula for Multiple Orthogonal Polynomials of Mixed Type

2013

An alternative expression for the Christoffel--Darboux formula for multiple orthogonal polynomials of mixed type is derived from the LULULU factorization of the moment matrix of a given measure and two sets of weights. We use the action of the generalized Jacobi matrix JJJ, also responsible for the recurrence relations, on the linear forms and their duals to obtain the result.

A Jacobi type Christoffel–Darboux formula for multiple orthogonal polynomials of mixed type

Linear Algebra and its Applications Volume 468, 1 March 2015, Pages 154–170 18th ILAS Conference

An alternative expression for the Christoffel–Darboux formula for multiple orthogonal polynomials of mixed type is derived from the LU factorization of the moment matrix of a given measure and two sets of weights. We use the action of the generalized Jacobi matrix J, also responsible for the recurrence relations, on the linear forms and their duals to obtain the result.

An algebraic theory about semiclassical and classical matrix orthogonal polynomials

In this paper we introduce an algebraic theory of classical matrix orthogonal polynomials as a particular case of the semi-classical ones, defined by a distributional equation for the corresponding orthogonality functional. This leads to several properties that characterize the classical matrix families, among them, a structure relation and a second order differo-differential equation. In the particular case of Hermite type matrix polynomials we obtain all the parameters associated with the family and we prove that they satisfy, not only a differo-differential equation, but a second order differential one, as it can be seen in the scalar case.

The symmetric Dunkl-classical orthogonal polynomials revisited

arXiv (Cornell University), 2024

We investigate the symmetric Dunkl-classical orthogonal polynomials by using a new approach applied in connection with the Dunkl operator. The main aim of this technique is to determine the recurrence coefficients first and foremost. We establish the existence and uniqueness of symmetric Dunkl-classical orthogonal polynomials, and also confirm that only two families of orthogonal polynomials, that is, the generalized Hermite polynomials and the generalized Gegenbauer polynomials, belong to this class. Two apparently new characterizations in a more general setting are given. This paper complements earlier work of Ben Cheikh and Gaied.

Properties of matrix orthogonal polynomials via their Riemann–Hilbert characterization

2011

We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder operators, some of them of 0-th order, something that is not possible in the scalar case. The combination of the ladder operators will lead to a family of second-order differential equations satisfied by the orthogonal polynomials, some of them of 0-th and first order, something also impossible in the scalar setting. This shows that the differential properties in the matrix case are much more complicated than in the scalar situation. We will study several examples given in the last years as well as others not considered so far.

On the associated orthogonal polynomials

Journal of Computational and Applied Mathematics, 1990

By using the second-order recurrence relation this paper gives some new results on associated orthogonal polynomials without referring to the continued fractions' tool. Some results are very useful for obtaining the second-order differential equation satisfied by the semi-classical orthogonal polynomials (Hendriksen and van Rossum (1985) Maroni (1987)) (cf. Section 3). Also, the main formula derives from Proposition 2.6, by which the fourth-order differential equation, satisfied by some Laguerre-Hahn polynomials (Magnus (1984)), is obtained (cf. Behnehdi and Ronveaux (1989), Dini et al. (1989), Ronveaux et al. (1990)).

A matrix Rodrigues formula for classical orthogonal polynomials in two variables

Journal of Approximation Theory, 2009

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.

Connection relations and characterizations of orthogonal polynomials

Advances in Applied Mathematics, 2012

We give a general method of characterizing symmetric orthogonal polynomials through a certain type of connection relations. This method is applied to Al-Salam-Chihara, Askey-Wilson, and Meixner-Pollaczek polynomials. This characterization technique unifies and extends some previous characterization results of Lasser and Obermaier and Ismail and Obermaier. Along the way we explicitly evaluate the connection coefficients in the expansion of D 2 q p n in terms of {p k }, where D q is the Askey-Wilson operator and {p k } are general Askey-Wilson polynomials. As a limiting case we derive the corresponding connection coefficients in the expansion of W 2 W n in terms of {W k }, where W is the Wilson operator and {W k } are general Wilson polynomials. Using the connection relation for Askey-Wilson polynomials, we obtain a characterization for the two-parameter symmetric Askey-Wilson polynomials. The connection relations between D m P (α,β) n , D := d/dx and {P (α,β) k } are also derived.