# q-Middle Convolution and q-Painlevé Equation

@article{Sasaki2022qMiddleCA, title={q-Middle Convolution and q-Painlev{\'e} Equation}, author={Shoko Sasaki and Shunsuke Takagi and Kouichi Takemura}, journal={Symmetry, Integrability and Geometry: Methods and Applications}, year={2022} }

A q-deformation of the middle convolution was introduced by Sakai and Yamaguchi. We apply it to a linear q-difference equation associated with the q-Painlevé VI equation. Then we obtain integral transformations. We investigate the q-middle convolution in terms of the affine Weyl group symmetry of the q-Painlevé VI equation. We deduce an integral transformation on the q-Heun equation.

## One Citation

### On q-middle convolution and q-hypergeometric equations

- Mathematics
- 2022

. We reformulate q -integral transformations associated with the q -middle convolution. We obtain q -integral representations of the variants of the q -hypergeometric equation by applying the q…

## References

SHOWING 1-10 OF 19 REFERENCES

### A q-analog of the sixth Painlevé equation

- Mathematics
- 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…

### On q-middle convolution and q-hypergeometric equations

- Mathematics
- 2022

. We reformulate q -integral transformations associated with the q -middle convolution. We obtain q -integral representations of the variants of the q -hypergeometric equation by applying the q…

### q-Heun equation and initial-value space of q-Painlev\'e equation

- Mathematics, Physics
- 2021

We show that the q-Heun equation and its variants appear in the linear q-difference equations associated to some q-Painlevé equations by considering the blow-up associated to their initial-value…

### Spectral types of linear $q$-difference equations and $q$-analog of middle convolution

- Mathematics
- 2014

We give a $q$-analog of middle convolution for linear $q$-difference equations with rational coefficients. In the differential case, middle convolution is defined by Katz, and he examined properties…

### Degenerations of Ruijsenaars-van Diejen operator and q-Painleve equations

- Mathematics
- 2016

It is known that the Painleve VI is obtained by connection preserving deformation of some linear differential equations, and the Heun equation is obtained by a specialization of the linear…

### On q-Deformations of the Heun Equation

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2018

The q-Heun equation and its variants were obtained by degenerations of Ruijsenaars-van Diejen operators with one particle. We investigate local properties of these equations. Especially we…

### Middle Convolution and Heun's Equation

- Mathematics
- 2009

Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of ini- tial conditions of the sixth…

### Middle convolution of Fuchsian systems and the construction of rigid differential systems

- Mathematics
- 2007

### Geometric aspects of Painlevé equations

- Mathematics
- 2015

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