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

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

2009-06-03

arxiv.org/abs/0906.0640v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

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

Scientific paper

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.

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