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Symmetry, Integrability and Geometry: Methods and Applications, 2011, Volume 7, 101, 54 pp.
DOI: https://doi.org/10.3842/SIGMA.2011.101
(Mi sigma659)
 

This article is cited in 8 scientific papers (total in 8 papers)

A Relativistic Conical Function and its Whittaker Limits

Simon Ruijsenaars

School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
Full-text PDF (803 kB) Citations (8)
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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.
Received: April 30, 2011; in final form October 23, 2011; Published online November 1, 2011
Bibliographic databases:
Document Type: Article
Language: English
Citation: Simon Ruijsenaars, “A Relativistic Conical Function and its Whittaker Limits”, SIGMA, 7 (2011), 101, 54 pp.
Citation in format AMSBIB
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\by Simon Ruijsenaars
\paper A Relativistic Conical Function and its Whittaker Limits
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\vol 7
\papernumber 101
\totalpages 54
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  • This publication is cited in the following 8 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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