A degenerate Gaussian weight connected with Painlevé equations and Heun equations

Abstract

In this paper, we study the recurrence coefficients of a deformed Hermite polynomials orthogonal with respect to the weight [Formula: see text] where [Formula: see text] and [Formula: see text]. It is an extension of Chen and Feigin [J. Phys. A., Math. Gen. 39(2006) 12381–12393]. By using the ladder operator technique, we show that the recurrence coefficients satisfy a particular Painlevé IV equation and the logarithmic derivative of the associated Hankel determinant satisfies the Jimbo–Miwa–Okamoto [Formula: see text] form of the Painlevé IV. Furthermore, the asymptotics of the recurrence coefficients and the Hankel determinant are obtained at the hard-edge limit and can be expressed in terms of the solutions to the Painlevé XXXIV and the [Formula: see text]-form of the Painlevé II equation at the soft-edge limit, respectively. In addition, for the special case [Formula: see text], we obtain the asymptotics of the Hankel determinant at the hard-edge limit via semi-classical Laguerre polynomials with respect to the weight [Formula: see text], which reproduced the result in Charlier and Deano, [Integr. Geom. Methods Appl. 14(2018) 018 (p. 43)].