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Product formulas for the relativistic and nonrelativistic conical functions

Conference paper
Authors Martin Hallnäs
Simon Ruijsenaars
Published in Representation theory, special functions and Painlevé equations. RIMS 2015, Kyoto, Japan, March 3-6, 2015, s. 195-245
Publisher Mathematical Society of Japan / World Scientific Publishing
Publication year 2018
Published at Department of Mathematical Sciences
Language en
Links https://doi.org/10.1142/e057
Keywords product formulas, quantum Calogero-Moser systems, conical function
Subject categories Mathematical Analysis, Control Engineering, Production Engineering, Human Work Science and Ergonomics

Abstract

The conical function and its relativistic generalization can be viewed as eigenfunctions of the reduced 2-particle Hamiltonians of the hyperbolic Calogero-Moser system and its relativistic generalization. We prove new product formulas for these functions. As a consequence, we arrive at explicit diagonalizations of integral operators that commute with the 2-particle Hamiltonians and reduced versions thereof. The kernels of the integral operators are expressed as integrals over products of the eigenfunctions and explicit weight functions. The nonrelativistic limits are controlled by invoking novel uniform limit estimates for the hyperbolic gamma function.

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