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

@inproceedings{Dzhamay2021DifferentialEF, title={Differential equations for the recurrence coefficients of semi-classical orthogonal polynomials and their relation to the Painlev\'e equations via the geometric approach}, author={Anton Dzhamay and Galina Filipuk and Alexander Stokes}, year={2021} }

Abstract In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painlevé equations using the geometric framework of the Okamoto Space of Initial Conditions. We demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing in [HC17], we show how the recurrence coefficients are connected to the fourth Painlevé equation. For discrete orthogonal polynomials…

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Different Hamiltonians for the Painlev\'e ${\text{P}_{\mathrm{IV}}}$ equation and their identification using a geometric approach

- Physics, Mathematics
- 2021

It is well-known that differential Painlevé equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique – there are many very different…

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