Abstract
In this paper, we study four kinds of polynomials: orthogonal with the singularly perturbed Gaussian weight wSPG(x), the deformed Freud weight wDF(x), the jumpy Gaussian weight wJG(x), and the Jacobi-type weight wJC(x). The second order linear differential equations satisfied by these orthogonal polynomials and the associated Heun equations are presented. Utilizing the method of isomonodromic deformations given by Dereziński et al. [Symmetry Integr. Geom. Methods Appl. 17, 056 (2021)], we transform these Heun equations into Painlevé equations. It is interesting that the Painlevé equations obtained by the way in this work are same as the results satisfied by the related three term recurrence coefficients or the auxiliaries studied by other authors. In addition, we discuss the asymptotic behaviors of the Hankel determinant generated by the first weight, wSPG(x), under a suitable double scaling for large s and small s, where the Dyson’s constant is recovered.
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