• DOI:10.3842/SIGMA.2019.074
• Corpus ID: 119314863

Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations

@article{Matsuhira2019CombinatorialEF,
  title={Combinatorial Expressions for the Tau Functions of q-Painlev{\'e} V and III Equations},
  author={Yuya Matsuhira and H. Nagoya},
  journal={Symmetry, Integrability and Geometry: Methods and Applications},
  year={2019}
}

• Y. Matsuhira, H. Nagoya
• Published 8 November 2018
• Mathematics
• Symmetry, Integrability and Geometry: Methods and Applications

We derive series representations for the tau functions of the $q$-Painlev\'e V, $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations, as degenerations of the tau functions of the $q$-Painlev\'e VI equation in [Jimbo M., Nagoya H., Sakai H., J. Integrable Syst. 2 (2017), xyx009, 27 pages]. Our tau functions are expressed in terms of $q$-Nekrasov functions. Thus, our series representations for the tau functions have explicit combinatorial structures. We show that general solutions to the $q$-Painlev\'e V, $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations are written by our tau functions. We also prove that our tau functions for the $q$-Painlev\'e $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations satisfy the three-term bilinear equations for them.