A high order qq-difference equation for qq-Hahn multiple orthogonal polynomials

A high order linear $q$-difference equation with polynomial coefficients having $q$-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation is related to the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when $q\to1$ are studied. Indeed, the difference equation for Hahn multiple orthogonal polynomials given in \cite{Lee} is corrected and obtained as a limiting case

Casorati Type Determinants Of Some Q-Classical Orthogonal Polynomials

Some symmetries for Casorati determinants whose entries are q-classical orthogonal polynomials are studied. Special attention is paid to the symmetry involving Big q-Jacobi polynomials. Some limiting situations, for other related q-classical orthogonal polynomial families in the q-Askey scheme, namely q-Meixner, q-Charlier, and q-Laguerre polynomials, are considered.This work was partially supported by MTM2012-36732-C03-03 (Ministerio de Economía y Competitividad), FQM-262, FQM-4643, FQM-7276 (Junta de Andalucía) and Feder Funds (European Union). The research of the second author was partially supported by the project MTM2012- 36732-C03-01 (Ministerio de Econom´ıa y Competitividad)

Some discrete multiple orthogonal polynomials

27 pages, no figures.-- MSC2000 codes: 33C45, 33C10, 42C05, 41A28.-- Issue title: "Proceedings of the 6th International Symposium on Orthogonal Polynomials, Special Functions and their Applications" (OPSFA-VI, Rome, Italy, 18-22 June 2001).MR#: MR1985676 (2004g:33015)Zbl#: Zbl 1021.33006In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn (T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978; R. Koekoek and R.F. Swarttouw, Reports of the Faculty of Technical Mathematics and Informatics No. 98-17, Delft, 1998; A.F. Nikiforov et al., Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991). These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order difference equation, and an explicit expression from which the coefficients of the three-term recurrence relation can be obtained. Then we consider r positive discrete measures and define two types of multiple orthogonal polynomials. The continuous case (Jacobi, Laguerre, Hermite, etc.) was studied by Van Assche and Coussement (J. Comput. Appl. Math. 127 (2001) 317–347) and Aptekarev et al. (Multiple orthogonal polynomials for classical weights, manuscript). The families of multiple orthogonal polynomials (of type II) that we will study have a raising operator and hence a Rodrigues formula. This will give us an explicit formula for the polynomials. Finally, there also exists a recurrence relation of order r+1 for these multiple orthogonal polynomials of type II. We compute the coefficients of the recurrence relation explicitly when r=2.This research was supported by INTAS project 00-272, Dirección General de Investigación del Ministerio de Ciencia y Tecnología of Spain under grants BFM-2000-0029 and BFM-2000-0206-C04-01, Dirección General de Investigación de la Comunidad Autónoma de Madrid, and by project G.0184.02 of FWO-Vlaanderen.Publicad

Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s(d + 1) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials.We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions.We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite- Padé approximation in matrix form is given