Multiple orthogonal polynomials: Pearson equations and Christoffel formulas

Abstract

Multiple orthogonal polynomials with respect to two weights on the step-line are considered. A connection between different dual spectral matrices, one banded (recursion matrix) and one Hessenberg, respectively, and the Gauss–Borel factorization of the moment matrix is given. It is shown a hidden freedom exhibited by the spectral system related to the multiple orthogonal polynomials. Pearson equations are discussed, a Laguerre–Freud matrix is considered, and differential equations for type I and II multiple orthogonal polynomials, as well as for the corresponding linear forms are given. The Jacobi–Piñeiro multiple orthogonal polynomials of type I and type II are used as an illustrating case and the corresponding differential relations are presented. A permuting Christoffel transformation is discussed, finding the connection between the different families of multiple orthogonal polynomials. The Jacobi–Piñeiro case provides a convenient illustration of these symmetries, giving linear relations between different polynomials with shifted and permuted parameters. We also present the general theory for the perturbation of each weight by a different polynomial or rational function aka called Christoffel and Geronimus transformations. The connections formulas between the type II multiple orthogonal polynomials, the type I linear forms, as well as the vector Stieltjes–Markov vector functions is also presented. We illustrate these findings by analyzing the special case of modification by an even polynomial.

Laguerre–Freud equations for three families of hypergeometric discrete orthogonal polynomials

“…[16] and study its relations with the transformations presented in Ref. [20] and quadrilateral lattices 23,45 ii)An important issue is the connection with discrete and continuous Painlevé equations for the coefficients $$\beta _n$$ and $$\gamma _n$$, taking into account the study of such a question when you deal with discrete semiclassical linear functionals given in Ref.…”

“…Let us assume a weight 𝑤 satisfying the Pearson equation (17). Then, the Laguerre-Freud structure matrix Ψ given in (20) satisfies…”

Riemann–Hilbert Problem and Matrix Biorthogonal Polynomials

“…In [6], we provided hypergeometric expressions for the Hahn multiple orthogonal polynomials of type I, which allowed for the derivation of explicit hypergeometric expressions for all type I multiple orthogonal polynomials that are descendants in the multiple Askey scheme of the Hahn multiple orthogonal polynomials. These descendants include Jacobi-Piñeiro, multiple Laguerre of the first and second kind, multiple Meixner of the first and second kind, multiple Charlier, and multiple Kravchuk.…”

Laguerre-Freud Equations for three families of hypergeometrical discrete orthogonal polynomials

“…ii) The Jacobi matrix satis es Π −1 Hθ(J ) = σ(J)HΠ , (19) and the matrices Hθ(J ) and σ(J)H are symmetric.…”

“…In [26] and this paper we have shown that is possible for the cases (p, q) = (0, 1), (1, 1), (2, 1), (1, 2), (2, 2), (3,2). For the future, we will also extend these techniques to multiple discrete orthogonal polynomials [15] and study its relations with the transformations presented in [19] and quadrilateral lattices [21,41]. R…”