Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Symmetry Integrability and Geometry-Methods and Applications

article

Mourad E.H. Ismail¹  Erik Koelink²  Pablo Román³ 

[1] University of Central Florida;Radboud Universiteit;Universidad Nacional de Córdoba, Medina Allende s/n Ciudad Universitaria

关键词: orthogonal polynomials;    Askey scheme and its q-analogue;    expansion formulas;    Toda lattice;   

DOI  :  10.3842/SIGMA.2018.072

来源: National Academy of Science of Ukraine

【 摘 要 】

Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big $q$-Jacobi polynomials and big $q$-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials.

【 授权许可】

Unknown