Linear spectral transformations for multivariate orthogonal polynomials and multispectral Toda hierarchies

Abstract

Linear spectral transformations of orthogonal polynomials in the real line, and in particular Geronimus transformations, are extended to orthogonal polynomials depending on several real variables. Multivariate Christoffel-Geronimus-Uvarov formulae for the perturbed orthogonal polynomials and their quasi-tau matrices are found for each perturbation of the original linear functional. These expressions are given in terms of quasi-determinants of bordered truncated block matrices and the 1D Christoffel-Geronimus-Uvarov formulae in terms of quotient of determinants of combinations of the original orthogonal polynomials and their Cauchy transforms, are recovered. A new multispectral Toda hierarchy of nonlinear partial differential equations, for which the multivariate orthogonal polynomials are reductions, is proposed. This new integrable hierachy is associated with non-standard multivariate biorthogonality. Wave and Baker functions, linear equations, Lax and Zakharov-Shabat equations, KP type equations, appropriate reductions, Darboux/linear spectral transformations, and bilinear equations involving linear spectral transformations are presented. Finally, the paper includes an Appendix devoted to multivariate Uvarov transformations. Particular attention is paid to 0D-Uvarov perturbations and also to the 1D-Uvarov perturbations, which require of the theory of Fredholm integral equations.

Multivariate Orthogonal Polynomials and Modified Moment Functionals

“…Also, as in the univariate case, they proved a connection with the Darboux factorization of the Jacobi block matrix associated with the three term recurrence relations for multivariate orthogonal polynomials. Similar considerations for multivariate Geronimus and more general linear spectral transformations of moment functionals can be found, among other topics, in [5].…”

“…The authors proved that, at least in the Legendre case, these multivariate orthogonal polynomials satisfy a fourth-order partial differential equation, which constitutes a natural extension of Krall orthogonal polynomials [16] to the multivariate case. In [5], a modification of a moment functional by adding another moment functional defined on a curve is presented, and a Christoffel formula built up in terms of a Fredholm integral equation is discussed. As far as we know, a general theory about Uvarov modifications by means of moment functionals defined on lower dimensional manifolds remains as an open problem.…”

“…In [7] the authors show that modifications of univariate moment functionals are related with the Darboux factorization of the associated Jacobi matrix. In this direction, in [3,4,5] long discussions about several aspects of the theory of multivariate orthogonal polynomials can be found. In particular, Darboux transformations for orthogonal polynomials in several variables were presented in [3], and in [4] we can find an extension of the univariate Christoffel determinantal formula to the multivariate context.…”

Christoffel Transformations for Matrix Orthogonal Polynomials in the Real Line and the non-Abelian 2D Toda Lattice Hierarchy

“…This first order linear ordinary differential system can be re-casted as a matrix linear differential equation as follows (19)…”

“…It is well known that there is a deep connection between discrete integrable systems and Darboux transformations of continuous integrable systems, see for example [41]. Finally, let us comment that, in the realm of several variables, in [17,18,19] one can find extensions of the Christoffel formula to the multivariate scenario with real variables and on the unit torus, respectively. 1.2.…”