Multiple orthogonal polynomials: Pearson equations and Christoffel formulas
| Autor: | Amílcar Branquinho, Manuel Manas, Ana Foulquié Moreno |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
Algebra and Number Theory
Física-Modelos matemáticos Multiple orthogonal polynomials Geronimus transformations FOS: Physical sciences Banded tetradiagonal recursion matrices Mathematical Physics (math-ph) Christoffel transformations Pearson equation Mathematics - Classical Analysis and ODEs Classical Analysis and ODEs (math.CA) FOS: Mathematics Física matemática 42C05 33C45 33C47 Analysis Mathematical Physics |
| Zdroj: | E-Prints Complutense. Archivo Institucional de la UCM instname |
| Popis: | 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-Pi\~neiro 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-Pi\~neiro 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. Comment: Revised version, completely new section on general Christoffel and Geronimus for multiple orthogonal polynomials on the stepline |
| Databáze: | OpenAIRE |
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