Painlevé V for a Jacobi unitary ensemble with random singularities. (English) Zbl 1485.34214
A special solution of the fifth Painlevé equation is represented by an integral of orthogonal polynomials.
The authors start from a perturbed Jacobi weight \[ w(z;t)=(1-z^2)^\alpha \exp \left[ - \frac{t}{z^2-k^2} \right], \quad k\in [-1,1], \] for \(\alpha, t>0\). Let \(P_n(z;t)\) be a monic orthogonal polynomials of degree \(n\) in \(z\) with respect to \(w(z;t)\): \[ \int_{-1}^1 P_n(z;t)P_m(z;t) w(z;t) \, dz =h_n(t)\delta_{n,m} \quad m, n = 0, 1, 2,\dots. \] By using the ladder operator approach, they show that an auxiliary function \[ R_n(t)= \frac{2t} {h_n(t) } \int_{-1}^1 \frac{1}{z^2 -k^2 }P_n(z;t)^2 w(z;t) \, dz \] satisfy a recurrence relation and \( \Phi_n(t)= R_n(t)/ ( 2n+ \alpha +1) +1 \) is a solution of a particular fifth Painlevé equation.
The authors start from a perturbed Jacobi weight \[ w(z;t)=(1-z^2)^\alpha \exp \left[ - \frac{t}{z^2-k^2} \right], \quad k\in [-1,1], \] for \(\alpha, t>0\). Let \(P_n(z;t)\) be a monic orthogonal polynomials of degree \(n\) in \(z\) with respect to \(w(z;t)\): \[ \int_{-1}^1 P_n(z;t)P_m(z;t) w(z;t) \, dz =h_n(t)\delta_{n,m} \quad m, n = 0, 1, 2,\dots. \] By using the ladder operator approach, they show that an auxiliary function \[ R_n(t)= \frac{2t} {h_n(t) } \int_{-1}^1 \frac{1}{z^2 -k^2 }P_n(z;t)^2 w(z;t) \, dz \] satisfy a recurrence relation and \( \Phi_n(t)= R_n(t)/ ( 2n+ \alpha +1) +1 \) is a solution of a particular fifth Painlevé equation.
Reviewer: Yousuke Ohyama (Tokushima)
MSC:
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
| 33C47 | Other special orthogonal polynomials and functions |
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