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In this paper we present a general scheme for how to relate differentialequations for the recurrence coefficients of semi-classical orthogonalpolynomials to the Painlev\'e equations using the geometric framework of theOkamoto Space of Initial Conditions. We demonstrate this procedure in twoexamples. For semi-classical Laguerre polynomials appearing in \cite{HC17}, weshow how the recurrence coefficients are connected to the fourth Painlev\'eequation. For discrete orthogonal polynomials associated with thehypergeometric weight appearing in \cite{FVA18} we discuss the relation of therecurrence coefficients to the sixth Painlev\'e equation, extending the resultsof \cite{DFS19}, where a similar approach was used for a discrete system forthe same recurrence coefficients. Though the discrete and differential systemshere share the same geometry, the construction of the space of initialconditions from the differential system is different and reveals extraconsiderations that must be made. We also discuss a number of related topics inthe context of the geometric approach, such as Hamiltonian forms of thedifferential equations for the recurrence coefficients, Riccati solutions forspecial parameter values, and associated discrete Painlev\'e equations.

Differential equations for the recurrence coefficients of semi-classical  orthogonal polynomials and their relation to the Painlev\'e equations via the  geometric approach