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arXiv · 1811.03285

Combinatorial expressions for the tau functions of $q$-Painlev\'e V and III equations

Yuya Matsuhira·Hajime Nagoya
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Abstract

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.

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Yuya Matsuhira, Hajime Nagoya. 2018-11-08. Combinatorial expressions for the tau functions of $q$-Painlev\'e V and III equations. https://doi.org/10.3842/sigma.2019.074

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