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

Abstract

We derive series representations for the tau functions of the q-Painlevé V, III₁, III₂, and III₃ equations, as degenerations of the tau functions of the q-Painlevé 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-Painlevé V, III₁, III₂, and III₃ equations are written by our tau functions. We also prove that our tau functions for the q-Painlevé III₁, III₂, and III₃ equations satisfy the three-term bilinear equations for them.