# Construction of a Lax Pair for the E6(1)q-Painlevé System

@article{Witte2012ConstructionOA, title={Construction of a Lax Pair for the E6(1)q-Painlev{\'e} System}, author={Nicholas S. Witte and Christopher M. Ormerod}, journal={Symmetry Integrability and Geometry-methods and Applications}, year={2012}, volume={8}, pages={097} }

We construct a Lax pair for the $E^{(1)}_6 $ $q$-Painleve system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices - the $q$-linear lattice - through a natural generalisation of the big $q$-Jacobi weight. As a by-product of our construction we derive the coupled first-order $q$-difference equations…

## 16 Citations

The Padé interpolation method applied to q-Painlevé equations II (differential grid version)

- Physics, Mathematics
- 2017

Recently, we studied Padé interpolation problems of q-grid, related to q-Painlevé equations of type $$E_7^{(1)}$$E7(1), $$E_6^{(1)}$$E6(1), $$D_5^{(1)}$$D5(1), $$A_4^{(1)}$$A4(1) and…

Discrete Painlev\'e equations and their Lax pairs as reductions of integrable lattice equations

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We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations…

The Padé Interpolation Method Applied to q-Painlevé Equations

- Mathematics, Physics
- 2015

We establish interpolation problems related to all the q-Painlevé equations of types from $${E_7^{(1)}}$$E7(1) to $${(A_2+A_1)^{(1)}}$$(A2+A1)(1). By solving those problems, we can derive the…

Geometric Aspects of Painlev\'e Equations

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In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev\'e equations, with a particular emphasis on the discrete Painlev\'e…

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We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations…

Hypergeometric Solutions for the q-Painlevé Equation of Type $${E^{(1)}_6}$$ by the Padé Method

- Mathematics, Physics
- 2013

The q-Painlevé equation of type $${E^{(1)}_6}$$ is obtained by the Padé method. Special solutions in a determinant formula to the q-Painlevé equations are presented. A relation between the Padé…

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- Mathematics, Physics
- 2015

Our goal is to show that the one-interval gap probability for the q-Hahn orthogonal polynomial ensemble can be expressed through a solution of the asymmetric q-Painleve V equation. The case of the…

Hypergeometric special solutions for $d$-Painlev\'e equations

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- 2017

We have studied Pade interpolation problems on an additive grid, related to additive difference ($d$-) Painleve equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$. By solving…

Lax pairs of discrete Painlevé equations: (A2+A1)(1) case

- Mathematics, PhysicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2016

In this paper, we provide a comprehensive method for constructing Lax pairs of discrete Painlevé equations by using a reduced hypercube structure. In particular, we consider the A5(1)-surface…

Hypergeometric special solutions to additive difference Painlevé equations by Padé method

- Mathematics
- 2017

We have studied Pad\'e interpolation problems on an additive grid, related to additive difference ($d$-) Painlev\'e equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$. By solving…

## References

SHOWING 1-10 OF 22 REFERENCES

Lax formalism for q-Painleve equations with affine Weyl group symmetry of type E^{(1)}_n

- Mathematics, Physics
- 2010

An explicit form of the Lax pair for the q-difference Painleve equation with affine Weyl group symmetry of type E^{(1)}_8 is obtained. Its degeneration to E^{(1)}_7, E^{(1)}_6 and D^{(1)}_5 cases are…

Lax form of the q-Painlevé equation associated with the A(1)2 surface

- Mathematics, Physics
- 2006

In this paper, the q-difference Painleve equation whose space of initial conditions is the A(1)2 surface is shown to appear as a particular case of a q-Garnier system, which is a q-difference counter…

Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

- Mathematics, PhysicsNagoya Mathematical Journal
- 2015

Abstract A 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials,…

A q-analog of the sixth Painlevé equation

- Mathematics, Physics
- 1996

A q-difference analog of the sixth Painlevé equation is presented. It arises as the condition for preserving the connection matrix of linear q-difference equations, in close analogy with the…

General theory of linear difference equations

- Mathematics
- 1911

The theory of lirlear diflerence equations with ratiollal coefficients was in a very backward state until POINCARG t in 1882 developed the notion of asymptotic representation, and its application to…

Lax forms of the q-Painlevé equations

- Physics, Mathematics
- 2008

All q-Painleve equations which are obtained from the q-analog of the sixth Painleve equation are expressed in a Lax formalism. They are characterized by the data of the associated linear q-difference…

Special function solutions of the discrete painlevé equations

- Mathematics
- 2001

In this paper, we present a review of the special solutions of discrete (difference) and q-discrete Painleve equations in terms of discrete special functions. These solutions exist whenever the…

Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials

- Mathematics
- 1988

One looks for [formal] orthogonal polynomials satisfying interesting differential or difference relations and equations (Laguerre-Hahn theory). The divided difference operator used here is…

Construction of hypergeometric solutions to the q-Painleve equations

- Mathematics, Physics
- 2005

Hypergeometric solutions to the q-Painlev\'e equations are constructed by direct linearization of disrcrete Riccati equations. The decoupling factors are explicitly determined so that the linear…

A q-Analog of the Garnier System

- Mathematics
- 2005

A q-difference analog of the Garnier system is presented. It arises as the condition for preserving the connection matrix of linear q-difference equations, in close analogy with the monodromy…