2015/11/29 by Gerardo Ariznabarreta, Ariznabarreta, Gerardo, Manuel Mañas +1
Chemistry · Mathematics · Physics and Astronomy · #14J70 #15A23 #33C45 #37K10 #37L60 #42C05 #46L55 #Classical Analysis and ODEs (math.CA) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1511.09129
openalex publication_date 2015/11/29 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Linear spectral transformations of orthogonal polynomials in the real line,\nand in particular Geronimus transformations, are extended to orthogonal\npolynomials depending on several real variables. Multivariate\nChristoffel-Geronimus-Uvarov formulae for the perturbed orthogonal polynomials\nand their quasi-tau matrices are found for each perturbation of the original\nlinear functional. These expressions are given in terms of quasi-determinants\nof bordered truncated block matrices and the 1D Christoffel-Geronimus-Uvarov\nformulae in terms of quotient of determinants of combinations of the original\northogonal polynomials and their Cauchy transforms, are recovered. A new\nmultispectral Toda hierarchy of nonlinear partial differential equations, for\nwhich the multivariate orthogonal polynomials are reductions, is proposed. This\nnew integrable hierachy is associated with non-standard multivariate\nbiorthogonality. Wave and Baker functions, linear equations, Lax and\nZakharov-Shabat equations, KP type equations, appropriate reductions,\nDarboux/linear spectral transformations, and bilinear equations involving\nlinear spectral transformations are presented.\n