The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation

Abstract

We show that the coefficients of the three-term recurrence relation for orthogonal polynomials with respect to a semi-classical extension of the Laguerre weight satisfy the fourth Painlevé equation when viewed as functions of one of the parameters in the weight. We compare different approaches to derive this result, namely, the ladder operators approach, the isomonodromy deformations approach and combining the Toda system for the recurrence coefficients with a discrete equation. We also discuss a relation between the recurrence coefficients for the Freud weight and the semi-classical Laguerre weight and show how it arises from the Bäcklund transformation of the fourth Painlevé equation.

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

“…Using Cauchy's integral formula, for ζ ∈ ∂U (0, ), we have 14) where the branch is chosen such that arg ζ ∈ (−π, π). A combination of (5.13) and (5.14) gives (5.12).…”

The generalized Krawtchouk polynomials and the fifth Painlevé equation

“…Journal of Difference Equations and Applications 1439 with initial conditions System (10) can be obtained by a limiting procedure from a-dP IV [15,32] given by…”

“…and letting 1 tend to zero, we get (10). It is worthwhile to point out that if we take a ¼ 0 and c ¼ p=ð1 -pÞ, the weight function (7) reduces to the one considered in [1], where the author derived a discrete system for the recurrence coefficients.…”

“…in the last two equations, we obtain system (10). Finally, we note that the initial conditions for the recurrence coefficients, in terms of the moments m 1 and m 0 , are given by a 2 0 ¼ 0 and…”

Ladder operators and differential equations for multiple orthogonal polynomials

“…On the other hand, ladder operators has been successfully applied to establish the connections between Painlevé equations and recurrence coefficients of certain orthogonal polynomials; cf. [6,9,13,14,19,21,22,23] for recent applications.…”