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In this paper we consider two examples of certain recurrence relations, ornonlinear discrete dynamical systems, that appear in the theory of orthogonalpolynomials, from the point of view of Sakai's geometric theory of Painlev\'eequations. On one hand, this gives us new examples of the appearance ofdiscrete Painlev\'e equations in the theory of orthogonal polynomials. On theother hand, it serves as a good illustration of the effectiveness of a recentlyproposed procedure on how to reduce such recurrences to some canonical discretePainlev\'e equations. Of particular interest is the fact that both recurrencesare regularized on the same family of rational algebraic surfaces, but at thesame time their dynamics are non-equivalent.

Recurrence relations for the generalized Laguerre and Charlier  orthogonal polynomials and discrete Painlev\'e equations on the $D_{6}^{(1)}$  Sakai surface