Abstract
We study the relation between semi-classical orthogonal polynomials and nonlinear differential equations coming from the isomonodromic deformation of linear system of differential equations on \(\mathbb{P}^{1}\). There are many works establishing this kind of relations between the Painlevé equations and semi-orthogonal polynomials with the weight functions taking from the integrands for hypergeometric, Kummer, Bessel, Hermite, Airy integrals. Some extension of these results is obtained for the semi-classical orthogonal polynomials with the weight functions coming from the general hypergeometric integrals on the Grassmannian G 2,N . To establish the desired relations, we make use of the Atiyah-Ward Ansatz construction of particular solutions for the 2 × 2 Schlesinger system and its degenerated ones.
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Acknowledgements
The author is supported in part by JSPS grants-in-aid No.15K04903.
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Kimura, H. (2017). Relation of Semi-Classical Orthogonal Polynomials to General Schlesinger Systems via Twistor Theory. In: Filipuk, G., Haraoka, Y., Michalik, S. (eds) Analytic, Algebraic and Geometric Aspects of Differential Equations. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-52842-7_12
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DOI: https://doi.org/10.1007/978-3-319-52842-7_12
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