Orthogonal polynomials, asymptotics, and Heun equations

Abstract

The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of “classical” weights multiplied by suitable “deformation factors,” usually dependent on a “time variable” t. From ladder operators [see A. Magnus, J. Comput. Appl. Math. 57(1-2), 215–237 (1995)], one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions. We will be interested in the situation when n, the order of the Hankel matrix and also the degree of the polynomials Pn(x) orthogonal with respect to the deformed weights, gets large. We show that the second order linear differential equations satisfied by Pn(x) are particular cases of Heun equations when n is large. In some sense, monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of Heun equations. Heun equations are of considerable importance in mathematical physics, and in the special cases, they degenerate to the hypergeometric and confluent hypergeometric equations. In this paper, we look at three types of weights: the Jacobi type, the Laguerre type, and the weights deformed by the indicator function of χ(a,b)(x) and the step function θ(x). In particular, we consider the following Jacobi type weights: (1.1) xα(1 − x)βe−tx, x ∈ [0, 1], α, β, t > 0; (1.2) xα(1 − x)βe−t/x, x ∈ (0, 1], α, β, t > 0; (1.3) (1−x2)α(1−k2x2)β, x∈[−1,1], α,β>0, k2∈(0,1); the Laguerre type weights: (2.1) xα(x + t)λe−x, x ∈ [0, ∞), t, α, λ > 0; (2.2) xαe−x−t/x, x ∈ (0, ∞), α, t > 0; and another type of deformation when the classical weights are multiplied by χ(a,b)(x) or θ(x): (3.1) e−x2(1−χ(−a,a)(x)), x∈R, a>0; (3.2) (1−x2)α(1−χ(−a,a)(x)), x∈[−1,1], a∈(0,1), α>0; (3.3) xαe−x(A + Bθ(x − t)), x ∈ [0, ∞), α, t > 0, A ≥ 0, A + B ≥ 0. The weights mentioned above were studied in a series of papers related to the deformation of “classical” weights.