Kernel functions and Bäcklund transformations for relativistic Calogero-Moser and Toda systems

Abstract

We obtain kernel functions associated with the quantum relativistic Toda systems, both for the periodic version and for the nonperiodic version with its dual. This involves taking limits of previously known results concerning kernel functions for the elliptic and hyperbolic relativistic Calogero-Moser systems. We show that the special kernel functions at issue admit a limit that yields generating functions of Bäcklund transformations for the classical relativistic Calogero-Moser and Toda systems. We also obtain the nonrelativistic counterparts of our results, which tie in with previous results in the literature.

76 pages

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• Explicit solution of the (quantum) elliptic Calogero-Sutherland model
• $(SL(N),q)$-opers, the $q$-Langlands correspondence, and quantum/classical duality
• On the N=2 superconformal index and eigenfunctions of the elliptic RS model
• Baxter operator and Baxter equation for $q$-Toda and Toda${}_2$ chains
• Action-angle map and duality for the open Toda lattice in the perspective of Hamiltonian reduction
• Integrable boundary interactions for Ruijsenaars' difference Toda chain
• Source identities and kernel functions for deformed (quantum) Ruijsenaars models
• On Razamat's $A_2$ and $A_3$ kernel identities
• Bispectral dual difference equations for the quantum Toda chain with boundary perturbations
• Stochastic Bäcklund transformations
• On R-matrix identities related to elliptic anisotropic spin Ruijsenaars-Macdonald operators
• Wave functions for quantum integrable particle systems via partial confluences of multivariate hypergeometric functions