Degenerations of Ruijsenaars-van Diejen operator and q-Painleve equations

Abstract

It is known that the Painleve VI is obtained by connection preserving deformation of some linear differential equations, and the Heun equation is obtained by a specialization of the linear differential equations. We inverstigate degenerations of the Ruijsenaars-van Diejen difference opearators and show difference analogues of the Painleve-Heun correspondence.

28 pages, references added, minor revision

References in corpus (7)

• Lax forms of the $q$-Painlevé equations
• A Lax Formalism for the Elliptic Difference Painlevé Equation
• Middle Convolution and Heun's Equation
• Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case
• Commutation Relations and Discrete Garnier Systems
• Lax formalism for q-Painleve equations with affine Weyl group symmetry of type E^{(1)}_n
• Heun equation and Painlevé equation

Cited by in corpus (16)

• On $q$-deformations of the Heun equation
• Spectral Theories and Topological Strings on del Pezzo Geometries
• Hanany-Witten Transition in Quantum Curves
• The Elliptic Painlevé Lax Equation vs. van Diejen's 8-Coupling Elliptic Hamiltonian
• Degenerate Sklyanin algebras, Askey-Wilson polynomials and Heun operators
• The rational Heun operator and Wilson biorthogonal functions
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• Variants of $q$-hypergeometric equation
• Elliptic Racah polynomials
• Polynomial solutions of $q$-Heun equation and ultradiscrete limit
• Ultradiscrete limit of the spectral polynomial of the $q$-Heun equation
• Variants of confluent q-hypergeometric equations
• Classical elliptic ${\mathrm{BC}}_1$ Ruijsenaars-van Diejen model: relation to Zhukovsky-Volterra gyrostat and 1-site classical XYZ model with boundaries
• q-Heun equation and initial-value space of q-Painlevé equation
• Heun's differential equation and its q-deformation
• Quantum Mirror Map for Del Pezzo Geometries