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, $\mathrm{II}{\mathrm{I}}_1$, $\mathrm{II}{\mathrm{I}}_2$, and $\mathrm{II}{\mathrm{I}}_3$ 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, $\mathrm{II}{\mathrm{I}}_1$, $\mathrm{II}{\mathrm{I}}_2$, and $\mathrm{II}{\mathrm{I}}_3$ equations are written by our tau functions. We also prove that our tau functions for the $q$-Painlevé $\mathrm{II}{\mathrm{I}}_1$, $\mathrm{II}{\mathrm{I}}_2$, and $\mathrm{II}{\mathrm{I}}_3$ equations satisfy the three-term bilinear equations for them.

References in corpus (6)

• Lax forms of the $q$-Painlevé equations
• Analyticity of Nekrasov Partition Functions
• Cluster Toda chains and Nekrasov functions
• CFT approach to the $q$-Painlevé VI equation
• Norm of the Whittaker vector of the deformed Virasoro algebra
• Remarks on irregular conformal blocks and Painlevé III and II tau functions

Cited by in corpus (7)

• A slow review of the AGT correspondence
• Quantum spectral problems and isomonodromic deformations
• Cluster integrable systems and spin chains
• Twisted Representations of Algebra of $q$-Difference Operators, Twisted $q$-$W$ Algebras and Conformal Blocks
• Folding transformations for q-Painleve equations
• On $q$-Isomonodromic Deformations and $q$-Nekrasov Functions
• Blowup relations on ${\mathbb{C}}^2/{\mathbb{Z}}_2$ from Nakajima-Yoshioka blowup relations