Connection preserving deformations and $q$-semi-classical orthogonal polynomials

Abstract

We present a framework for the study of $q$-difference equations satisfied by $q$-semi-classical orthogonal systems. As an example, we identify the $q$-difference equation satisfied by a deformed version of the little $q$-Jacobi polynomials as a gauge transformation of a special case of the associated linear problem for $q$-${\mathrm{P}}_{\mathrm{VI}}$. We obtain a parameterization of the associated linear problem in terms of orthogonal polynomial variables and find the relation between this parameterization and that of Jimbo and Sakai.

32 Pages, extended the study of the recursion relations, more references

References in corpus (2)

• Ladder Operators for q-orthogonal Polynomials
• The Symmetrical $H_q$-Semiclassical Orthogonal Polynomials of Class One

Cited by in corpus (2)

• Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy
• A study of the associated linear problem for $q$-${\mathrm{P}}_{\mathrm{V}}$