Differential equations for the recurrence coefficients of semiclassical orthogonal polynomials and their relation to the Painlevé equations via the geometric approach

We consider a general Sobolev-type inner product involving the Hahn difference operator, so this includes the well-known difference operators Dq and Δ and, as a limit case, the derivative operator. The objective is to construct the ladder operators for the corresponding nonstandard orthogonal polynomials and deduce the second-order differential-difference equation satisfied by these polynomials. Moreover, we will show that all the functions involved in these constructions can be computed explicitly.