# 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)

- 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

- Mathematics
- 2012

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

- Mathematics
- 2015

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…

On Lax representations of reductions of integrable lattice equations

- Mathematics
- 2012

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…

Moduli spaces of q-connections and gap probabilities

- Mathematics
- 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

- Mathematics
- 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

- MathematicsProceedings 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…

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