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A $\mathbb{D}$-semi-classical weight is one which satisfies a particularlinear, first order homogeneous equation in a divided-difference operator$\mathbb{D}$. It is known that the system of polynomials, orthogonal withrespect to this weight, and the associated functions satisfy a linear, firstorder homogeneous matrix equation in the divided-difference operator termed thespectral equation. Attached to the spectral equation is a structure whichconstitutes a number of relations such as those arising from compatibility withthe three-term recurrence relation. Here this structure is elucidated in thegeneral case of quadratic lattices. The simplest examples of the$\mathbb{D}$-semi-classical orthogonal polynomial systems are precisely thosein the Askey table of hypergeometric and basic hypergeometric orthogonalpolynomials. However within the $\mathbb{D}$-semi-classical class it isentirely natural to define a generalisation of the Askey table weights whichinvolve a deformation with respect to new deformation variables. We completelyconstruct the analogous structures arising from such deformations and theirrelations with the other elements of the theory. As an example we treat thefirst non-trivial deformation of the Askey-Wilson orthogonal polynomial systemdefined by the $q$-quadratic divided-difference operator, the Askey-Wilsonoperator, and derive the coupled first order divided-difference equationscharacterising its evolution in the deformation variable. We show that thissystem is a member of a sequence of classical solutions to the $ E^{(1)}_7 $$q$-Painlev\'e system.

Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy