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Deformation of the Askey-Wilson polynomials

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Discrete Integrable Systems

A $mathbb{D}$-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided difference operator $mathbb{D}$. 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. The simplest examples of the $mathbb{D}$-semi-classical orthogonal polynomial systems are precisely those in the Askey Table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the $mathbb{D}$-semi-classical class it is entirely natural to define a generalisation of the Askey Table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from the deformations and their relations with the other elements of the theory. As an example we treat the first non-trivial deformation of the system defined by the highest divided difference operator, the Askey-Wilson operator, that is to say the Askey-Wilson polynomials.

This talk is part of the Isaac Newton Institute Seminar Series series.

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