# Hermite-Padé approximation and integrability

@article{Doliwa2022HermitePadAA, title={Hermite-Pad{\'e} approximation and integrability}, author={Adam Doliwa and Artur Siemaszko}, journal={ArXiv}, year={2022}, volume={abs/2201.06829} }

We show that solution to the Hermite–Padé type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev–Petviashvili) system and of its adjoint linear problem. Our result explains the appearence of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorthms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite–Padé approximation…

## One Citation

### Integrability and geometry of the Wynn recurrence

- MathematicsNumerical Algorithms
- 2022

It is shown that the Wynn recurrence can be incorporated into the theory of integrable systems as a reduction of the discrete Schwarzian Kadomtsev–Petviashvili equation, allowing the geometric meaning of the recurrence to be presented as a construction of the appropriately constrained quadrangular set of points.

## References

SHOWING 1-10 OF 96 REFERENCES

### Discrete integrable systems generated by Hermite-Padé approximants

- Mathematics, Computer Science
- 2016

It is shown that the concept of multiple orthogonality is intimately related to the Lax representations for the entries of the nearest neighbor recurrence relations and it thus gives rise to a discrete integrable system.

### Hermite–Padé approximation, isomonodromic deformation and hypergeometric integral

- Mathematics
- 2015

We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite’s two approximation problems for…

### Nonlinear Methods in Numerical Analysis

- Mathematics
- 1987

I. Continued Fractions. Since these play an important role, the first chapter introduces their basic properties, evaluation algorithms and convergence theorems. From the section dealing with…

### Desargues maps and their reductions

- Mathematics
- 2013

We present recent developments on geometric theory of the Hirota system and of the non-commutative discrete Kadomtsev–Petviashvili (KP) hierarchy adding also some new results which make the picture…

### Generalized Orthogonal Polynomials, Discrete KP and Riemann–Hilbert Problems

- Mathematics
- 1999

Abstract:Classically, a single weight on an interval of the real line leads to moments, orthogonal polynomials and tridiagonal matrices. Appropriately deforming this weight with times t= (t1, t2, …),…

### Desargues maps and the Hirota–Miwa equation

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2009

We study the Desargues maps , which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multi-dimensional compatibility of the map is…

### Mixed type Hermite-Padé approximation inspired by the Degasperis-Procesi equation

- MathematicsAdvances in Mathematics
- 2019

### Do integrable mappings have the Painlevé property?

- MathematicsPhysical review letters
- 1991

We present an integrability criterion for discrete-time systems that is the equivalent of the Painlev\'e property for systems of a continuous variable. It is based on the observation that for…