Differential system related to generalised Krawtchouk polynomials: iterated regularisation and Painlevé equation

In this paper we tackle the generalised Krawtchouk polynomials. We show a straightforward connection between certain auxiliary quantities involving the recurrence coefficients of these polynomials and the fifth Painlevé equation via iterated regularisation. This connection follows an algorithmic procedure avoiding the use of less standardised methods. We further explore how to obtain polynomial systems via iterated regularisation and find decomposition of certain birational transformations into the product of simple ones.