Orthogonal polynomials with discontinuous weights

TL;DR

This paper explores a ladder operator approach to orthogonal polynomials with discontinuous weights, deriving difference and Toda equations, and connecting to Painleve IV for Hermite weights.

Contribution

It introduces a ladder operator formalism for orthogonal polynomials with discontinuous weights and derives new difference and Toda equations, linking to Painleve IV.

Findings

Derived difference equations for recurrence coefficients.

Established Toda evolution equations for recurrence coefficients.

Connected the equations to Painleve IV.

Abstract

In this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A_n(z) and B_n(z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w_0 and a standard jump function, then A_n(z) and B_n(z) have apparent simple poles at these jumps. We exemplify the approach by taking w_0 to be the Hermite weight. For this simpler case we derive, without using the compatibility conditions, a pair of difference equations satisfied by the diagonal and off-diagonal recurrence coefficients for a fixed location of the jump. We also derive a pair of Toda evolution equations for the recurrence coefficients which, when combined with the difference equations, yields a particular Painleve IV.