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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 − x 2 ) α (1 − k 2 x 2 ) β , x ∈ [ − 1 , 1 ] , α , β > 0 , k 2 ∈ (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 − x 2 (1 − χ (− a , a) (x)) , x ∈ R , a > 0 ; (3.2) (1 − x 2 ) 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. [ABSTRACT FROM AUTHOR]

In this paper, we study the probability density function, P (c , α , β , n) d c , of the center of mass of the finite n Jacobi unitary ensembles with parameters α > −1 and β > −1; that is the probability that trMn ∈ (c, c + dc), where Mn are n × n matrices drawn from the unitary Jacobi ensembles. We compute the exponential moment generating function of the linear statistics ∑ j = 1 n f ( x j ) ≔ ∑ j = 1 n x j , denoted by M f (λ , α , β , n). The weight function associated with the Jacobi unitary ensembles reads xα(1 − x)β, x ∈ [0, 1]. The moment generating function is the n × n Hankel determinant Dn(λ, α, β) generated by the time-evolved Jacobi weight, namely, w (x; λ, α, β) = xα(1 − x)β e−λx, x ∈ [0, 1], α > −1, β > −1. We think of λ as the time variable in the resulting Toda equations. The non-classical polynomials defined by the monomial expansion, Pn(x, λ) = xn + p(n, λ) xn−1 + ⋯ + Pn(0, λ), orthogonal with respect to w (x, λ, α, β) over [0, 1] play an important role. Taking the time evolution problem studied in Basor et al. [J. Phys. A: Math. Theor. 43, 015204 (2010)], with some change of variables, we obtain a certain auxiliary variable rn(λ), defined by integral over [0, 1] of the product of the unconventional orthogonal polynomials of degree n and n − 1 and w (x; λ, α, β)/x. It is shown that rn(2iez) satisfies a Chazy II equation. There is another auxiliary variable, denoted as Rn(λ), defined by an integral over [0, 1] of the product of two polynomials of degree n multiplied by w (x; λ, α, β)/x. Then Yn(−λ) = 1 − λ/Rn(λ) satisfies a particular Painlevé V: PV(α2/2, − β2/2, 2n + α + β + 1, 1/2). The σn function defined in terms of the λp(n, −λ) plus a translation in λ is the Jimbo–Miwa–Okamoto σ-form of Painlevé V. The continuum approximation, treating the collection of eigenvalues as a charged fluid as in the Dyson Coulomb Fluid, gives an approximation for the moment generating function M f (λ , α , β , n) when n is sufficiently large. Furthermore, we deduce a new expression of M f (λ , α , β , n) when n is finite, in terms the σ function of this is a particular case of Painlevé V. An estimate shows that the moment generating function is a function of exponential type and of order n. From the Paley-Wiener theorem, one deduces that P (c , α , β , n) has compact support [0, n]. This result is easily extended to the β ensembles, as long as the weight w is positive and continuous over [0, 1]. [ABSTRACT FROM AUTHOR]