Characterization of the Dᵂ-Laguerre-Hahn functionals

29 pages, no figures.-- MSC2000 codes: 33C45, 39A10.MR#: MR1914598 (2003e:33021)Zbl#: Zbl 1021.33007We give some characterization theorems for the DᵂLaguerre-Hahn linear functionals and we extend the concept of the class of the usual Laguerre-Hahn functionals to the Dᵂ-Laguerre-Hahn functionals, recovering the classic results when ᵂ tends to zero. Moreover, we show that some transformations carried out on the Dᵂ-Laguerre-Hahn linear functionals lead to new Dᵂ-Laguerre-Hahn linear functionals. Finally, we analyze the class of the resulting functionals and we give some applications relative to the first associated Charlier, Meixner, Krawtchouk and Hahn orthogonal polynomials.The work of the second author (FM) was supported by Ministerio de Ciencia y Tecnología (Dirección General de Investigación) of Spain under grant BFM 2000-0206-C04-01 and the INTAS project INTAS 2000-272.Publicad

Linear partial divided-difference equation satisfied by multivariate orthogonal polynomials on quadratic lattices

In this paper, a fourth-order partial divided-difference equation on quadratic lattices with polynomial coefficients satisfied by bivariate Racah polynomials is presented. From this equation we obtain explicitly the matrix coefficients appearing in the three-term recurrence relations satisfied by any bivariate orthogonal polynomial solution of the equation. In particular, we provide explicit expressions for the matrices in the three-term recurrence relations satisfied by the bivariate Racah polynomials introduced by Tratnik. Moreover, we present the family of monic bivariate Racah polynomials defined from the three-term recurrence relations they satisfy, and we solve the connection problem between two different families of bivariate Racah polynomials. These results are then applied to other families of bivariate orthogonal polynomials, namely the bivariate Wilson, continuous dual Hahn and continuous Hahn, the latter two through limiting processes. The fourth-order partial divided-difference equations on quadratic lattices are shown to be of hypergeometric type in the sense that the divided-difference derivatives of solutions are themselves solution of the same type of divided-difference equations.Comment: 36 page

Fourth-order difference equation satisfied by the co-recursive of q-classical orthogonal polynomials

AbstractWe derive the fourth-order q-difference equation satisfied by the co-recursive of q-classical orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials φ and ψ appearing in the q-Pearson difference equation Dq(φρ)=ψρ defining the weight ρ of the q-classical orthogonal polynomials inside the q-Hahn tableau. Use of suitable change of variable and limit processes allow us to recover the results known for the co-recursive of the classical continuous and classical discrete orthogonal polynomials. Moreover, we describe particular situations for which the co-recursive of classical orthogonal polynomials are still classical and express these new families in terms of the starting ones

Fourth Order q-Difference Equation for the First Associated of the q-Classical Orthogonal Polynomials

We derive the fourth order q-difference equation satisfied by the first associated of the q-classical orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials oe and ø which appear in the q-Pearson difference equation D q (oe ae) = ø ae defining the weight ae of the q-classical orthogonal polynomials inside the q-Hahn tableau. Keywords: q-Orthogonal polynomials, Fourth order q-difference equation. 1991 MSC: 33C25 1 Introduction The fourth order difference equation for the associated polynomials of all classical discrete polynomials were given for all integers r (order of association) in [5], using the properties of the Stieltjes functions of the associated linear forms. On the other hand, the equation for the first associated (r = 1) of all classical discrete polynomials was obtained in [13] using a useful relation proved in [2]. In this work, mimicking the approach used in [13] we give a single fourth order q-difference equation which is vali..

Laguerre-Freud equations for the recurrence coefficients of Dw semi-classical orthogonal polynomials of class one

AbstractThe Laguerre-Freud equations giving the recurrence coefficients βn, γn of orthogonal polynomials with respect to a Dω semi-classical linear form are derived. Dω is the difference operator. The limit when ω → 0 are also investigated recovering known results. Applications to generalized Meixner polynomials of class one are also treated