Orthogonal polynomial interpretation of Δ-Toda equations

• Autor(a) principal: Area, Iván
• Data de Publicação: 2015
• Outros Autores: Branquinho, Amílcar, Foulquié Moreno, Ana, Godoy, Eduardo
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• DOI: 10.1088/1751-8113/48/40/405206
• Texto Completo: http://hdl.handle.net/10316/43945; https://doi.org/10.1088/1751-8113/48/40/405206

In this paper a discretization of Toda equations is analyzed. The correspondence between these Δ-Toda equations for the coefficients of the Jacobi operator and its resolvent function is established. It is shown that the spectral measure of these operators evolve in t like ${(1+x)}^{1-t}\;{\rm{d}}\mu (x)$ where ${\rm{d}}\mu$ is a given positive Borel measure. The Lax pair for the Δ-Toda equations is derived and characterized in terms of linear functionals, where orthogonal polynomials which satisfy an Appell condition with respect to the forward difference operator Δ appear in a natural way. In order to illustrate the results of the paper we work out two examples of Δ-Toda equations related with Jacobi and Laguerre orthogonal polynomials.