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 SIGMA, 2011, Volume 7, 101, 54 pages (Mi sigma659)

A Relativistic Conical Function and its Whittaker Limits

Simon Ruijsenaars

School of Mathematics, University of Leeds, Leeds LS2 9JT, UK

Abstract: In previous work we introduced and studied a function $R(a_{+},a_{-},c;v,\hat{v})$ that is a generalization of the hypergeometric function $_2F_1$ and the Askey–Wilson polynomials. When the coupling vector $c\in\mathbb C^4$ is specialized to $(b,0,0,0)$, $b\in\mathbb C$, we obtain a function $\mathcal R (a_{+},a_{-},b;v,2\hat{v})$ that generalizes the conical function specialization of $_2F_1$ and the $q$-Gegenbauer polynomials. The function $\mathcal R$ is the joint eigenfunction of four analytic difference operators associated with the relativistic Calogero–Moser system of $A_1$ type, whereas the function $R$ corresponds to $BC_1$, and is the joint eigenfunction of four hyperbolic Askey–Wilson type difference operators. We show that the $\mathcal R$-function admits five novel integral representations that involve only four hyperbolic gamma functions and plane waves. Taking their nonrelativistic limit, we arrive at four representations of the conical function. We also show that a limit procedure leads to two commuting relativistic Toda Hamiltonians and two commuting dual Toda Hamiltonians, and that a similarity transform of the function $\mathcal R$ converges to a joint eigenfunction of the latter four difference operators.

Keywords: relativistic Calogero–Moser system, relativistic Toda system, relativistic conical function, relativistic Whittaker function.

DOI: https://doi.org/10.3842/SIGMA.2011.101

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Bibliographic databases:

ArXiv: 0807.0258
MSC: 33C05; 33E30; 39A10; 81Q05; 81Q80
Received: April 30, 2011; in final form October 23, 2011; Published online November 1, 2011
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Citation: Simon Ruijsenaars, “A Relativistic Conical Function and its Whittaker Limits”, SIGMA, 7 (2011), 101, 54 pp.

Citation in format AMSBIB
\Bibitem{Rui11} \by Simon Ruijsenaars \paper A Relativistic Conical Function and its Whittaker Limits \jour SIGMA \yr 2011 \vol 7 \papernumber 101 \totalpages 54 \mathnet{http://mi.mathnet.ru/sigma659} \crossref{https://doi.org/10.3842/SIGMA.2011.101} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000296884200001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84857081679} 

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This publication is cited in the following articles:
1. Hallnaes M. Ruijsenaars S., “Kernel Functions and Backlund Transformations for Relativistic Calogero–Moser and Toda Systems”, J. Math. Phys., 53:12 (2012), 123512
2. Hallnaes M. Ruijsenaars S., “Joint Eigenfunctions For the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type: i. First Steps”, Int. Math. Res. Notices, 2014, no. 16, 4400–4456
3. Hallnaes M., Ruijsenaars S., “a Recursive Construction of Joint Eigenfunctions For the Hyperbolic Nonrelativistic Calogero–Moser Hamiltonians”, Int. Math. Res. Notices, 2015, no. 20, 10278–10313
4. Hallnas M., Ruijsenaars S., “Joint Eigenfunctions For the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type II. the Two-and Three-Variable Cases”, Int. Math. Res. Notices, 2018, no. 14, 4404–4449
5. Hallnas M., Ruijsenaars S., “Product Formulas For the Relativistic and Nonrelativistic Conical Functions”, Representation Theory, Special Functions and Painleve Equations - Rims 2015, Advanced Studies in Pure Mathematics, 76, eds. Konno H., Sakai H., Shiraishi J., Suzuki T., Yamada Y., Math Soc Japan, 2018, 195–245
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