Differential system related to Krawtchouk polynomials: iterated regularisation and Painlev\'e equation

TL;DR

This paper investigates the regularisation of a differential system linked to Krawtchouk polynomials, revealing connections to Painlevé V and enabling polynomial systems and birational transformation decompositions.

Contribution

It introduces a novel iterative regularisation method connecting Krawtchouk polynomial coefficients to Painlevé V and explores polynomial systems and transformations.

Findings

Connection established between recurrence coefficients and Painlevé V.

Iterative regularisation yields polynomial systems.

Decompositions of birational transformations are achieved.

Abstract

We tackle the regularisation of a differential system related to generalised Krawtchouk polynomials. We show a straightforward connection between certain auxiliary quantities involving the recurrence coefficients of these polynomials and Painlev\'e V equation via iterative regularisation. Furthermore, we explore how iterative regularisation yields polynomial systems and enables us to find decompositions of certain birational transformations.