Two-Parameter Generalizations of Cauchy Bi-Orthogonal Polynomials and Integrable Lattices

In this article, we consider the generalised two-parameter Cauchy two-matrix model and the corresponding integrable lattice equation. It is shown that with parameters chosen as 1 / k i , k i ∈ Z > 0 ( i = 1 , 2 ), the average characteristic polynomials admit ( k 1 + k 2 + 2 ) -term recurrence rel...

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Bibliographic Details
Published in:Journal of nonlinear science 2021-04, Vol.31 (2), Article 30
Main Authors: Chang, Xiang-Ke, Li, Shi-Hao, Tsujimoto, Satoshi, Yu, Guo-Fu
Format: Article
Language:English
Subjects:
Analysis
Classical Mechanics
Economic Theory/Quantitative Economics/Mathematical Methods
Lattices
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical models
Mathematics
Mathematics and Statistics
Parameters
Partitions (mathematics)
Polynomials
Theoretical
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Summary:In this article, we consider the generalised two-parameter Cauchy two-matrix model and the corresponding integrable lattice equation. It is shown that with parameters chosen as 1 / k i , k i ∈ Z > 0 ( i = 1 , 2 ), the average characteristic polynomials admit ( k 1 + k 2 + 2 ) -term recurrence relations, which can be interpreted as spectral problems for integrable lattices. The tau function is then given by the partition function of the generalised Cauchy two-matrix model as well as Gram determinant. The simplest solvable example is given.
ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-021-09690-9