Nagoya Mathematical Journal

Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

N. S. Witte

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Abstract

A D-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator D. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the D-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the D-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the E7(1) q-Painlevé system.

Article information

Source
Nagoya Math. J., Volume 219 (2015), 127-234.

Dates
Received: 9 May 2012
Accepted: 31 October 2014
First available in Project Euclid: 20 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1445345520

Digital Object Identifier
doi:10.1215/00277630-3140952

Mathematical Reviews number (MathSciNet)
MR3413576

Zentralblatt MATH identifier
1334.39024

Subjects
Primary: 39A05: General theory
Secondary: 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45] 34M55: Painlevé and other special equations; classification, hierarchies; 34M56: Isomonodromic deformations 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 37K35: Lie-Bäcklund and other transformations

Keywords
nonuniform lattices divided-difference operators orthogonal polynomials semiclassical weights isomonodromic deformations Askey table Askey-Wilson polynomials

Citation

Witte, N. S. Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy. Nagoya Math. J. 219 (2015), 127--234. doi:10.1215/00277630-3140952. https://projecteuclid.org/euclid.nmj/1445345520


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