Product formulas for the relativistic and nonrelativistic conical functions

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.

Article information

Dates
First available in Project Euclid: 21 September 2018

https://projecteuclid.org/ euclid.aspm/1537499427

Digital Object Identifier
doi:10.2969/aspm/07610195

Mathematical Reviews number (MathSciNet)
MR3837923

Zentralblatt MATH identifier
07039304

Citation

Hallnäs, Martin; Ruijsenaars, Simon. Product formulas for the relativistic and nonrelativistic conical functions. Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, 195--245, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07610195. https://projecteuclid.org/euclid.aspm/1537499427