## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, H. Konno, H. Sakai, J. Shiraishi, T. Suzuki and Y. Yamada, eds. (Tokyo: Mathematical Society of Japan, 2018), 195 - 245

### Product formulas for the relativistic and nonrelativistic conical functions

Martin Hallnäs and Simon Ruijsenaars

#### 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**

Received: 30 August 2015

First available in Project Euclid:
21 September 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/aspm/07610195

**Mathematical Reviews number (MathSciNet)**

MR3837923

**Zentralblatt MATH identifier**

07039304

**Subjects**

Primary: 33C05: Classical hypergeometric functions, $_2F_1$ 33E30: Other functions coming from differential, difference and integral equations 39A70: Difference operators [See also 47B39] 47G10: Integral operators [See also 45P05] 81R12: Relations with integrable systems [See also 17Bxx, 37J35]

**Keywords**

product formulas conical function quantum Calogero-Moser systems

#### 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