Cauchy-Jacobi orthogonal polynomials and the discrete CKP equation

TL;DR

The paper develops a two-parameter Jacobi deformation of Cauchy bi-orthogonal polynomials and introduces discrete spectral transformations in the $s$ and $t$ directions. The normalization factor $\tau_n^{s,t}$ serves as a $τ$-function for the discrete CKP equation, and the authors derive a Lax-type framework and an integrable hierarchy from a four-term recurrence. They obtain a bilinear (and determinant) representation of the discrete CKP equation, with multiple-integral and determinant solutions for $τ_n^{s,t}$. This work connects Cauchy bi-orthogonal polynomials to discrete integrable systems and places the discrete CKP equation in a determinant/τ-function setting related to Yang-Baxter structures, integrable geometry, and cluster algebra theory.

Abstract

This paper intends to construct discrete spectral transformations for Cauchy-Jacobi orthogonal polynomials, and find its corresponding discrete integrable systems. It turns out that the normalization factor of Cauchy-Jacobi orthogonal polynomials acts as the $τ$-function of the discrete CKP equation, which has applications in Yang-Baxter equation, integrable geometry, cluster algebra, and so on.

Theorems & Definitions (24)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Remark 2.4
• Proposition 2.5
• Proposition 2.6
• proof
• Theorem 2.7
• Proposition 2.8
• proof