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

Galina Filipuk, Juan F. Mañas-Mañas, Juan J. Moreno-Balcázar, Cristina Rodríguez-Perales

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 Painlevé V equation via iterative regularisation. Furthermore, we explore how iterative regularisation yields polynomial systems and enables us to find decompositions of certain birational transformations.

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

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 Painlevé V equation via iterative regularisation. Furthermore, we explore how iterative regularisation yields polynomial systems and enables us to find decompositions of certain birational transformations.

Paper Structure

This paper contains 7 sections, 6 theorems, 123 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The differential system
  3. Regularisation of the system
  4. Iterated regularisation
  5. Connection to the fifth Painlevé equation
  6. Iterated regularisation for obtaining polynomial right--hand side systems
  7. Hamiltonian expressions

Key Result

Theorem 1

Krawtchouk Polynomials Let $a_n^2$ and $b_n$ be the recurrence coefficients in three-term generalised for the weight weight. The following quantities satisfy the discrete system with initial conditions where $M(a,b,z)$ is the confluent hypergeometric function $_1F_1(a;b;z)$ defined by (hypergeometric f).

Figures (3)

  • Figure 1: Diagram of the regularisation algorithm.
  • Figure 2: Diagram of the iterated regularisation algorithm.
  • Figure 3: Flow diagram.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 1 more