Orthogonal Polynomials for the Gaussian Weight with a Jump and Discrete Painlev\'e Equations

TL;DR

This paper refines the identification of discrete Painlevé equations associated with orthogonal polynomials for Gaussian weights with jumps, highlighting the importance of detailed dynamic descriptions beyond surface types.

Contribution

It introduces a refined approach to identifying discrete Painlevé equations, applied to a recurrence relation for orthogonal polynomials with a jump in the Gaussian weight.

Findings

Enhanced understanding of discrete Painlevé dynamics

Application to orthogonal polynomials with jumps

Improved identification methodology

Abstract

We describe a refined version of the discrete Painlev\'e identification problem that emphasizes the importance on going beyond just the surface type in describing a discrete Painlev\'e dynamic. We give an example of solving such identification problem for a particular recurrence relation that previously appeared in studying orthogonal polynomials for the Gaussian weight with a single jump.