### Abstract

In this paper, we present a new method of deducing infinite sequences of exact solutions of q-discrete Painleve equations by using their associated linear problems. The specific equation we consider in this paper is a q-discrete version of the second Painleve equation (q-PII) with affine Weyl group symmetry of type (A2 + A1)^{(1)}. We show, for the first time, how to use the q-discrete linear problem associated with q-PII to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q-PII here, is also applicable to other discrete Painlevé equations.

Original language | English |
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Pages (from-to) | 3443-3468 |

Number of pages | 26 |

Journal | PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES |

Volume | 467 |

Issue number | 2136 |

DOIs | |

Publication status | Published - 20 Jul 2011 |

Externally published | Yes |

### Keywords

- q-discrete
- Painlevé equations
- discrete equations

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## Cite this

*PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*,

*467*(2136), 3443-3468. https://doi.org/10.1098/rspa.2011.0167