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Elliptic Littlewood identities

Rains, Eric M. (2012) Elliptic Littlewood identities. Journal of Combinatorial Theory. Series A, 119 (7). pp. 1558-1609. ISSN 0097-3165. doi:10.1016/j.jcta.2012.03.001.

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We prove analogues for elliptic interpolation functions of Macdonaldʼs version of the Littlewood identity for (skew) Macdonald polynomials, in the process developing an interpretation of general elliptic “hypergeometric” sums as skew interpolation functions. One such analogue has an interpretation as a “vanishing integral”, generalizing a result of Rains and Vazirani (2007) [17]; the structure of this analogue gives sufficient insight to enable us to conjecture elliptic versions of most of the other vanishing integrals of Rains and Vazirani (2007) [17] as well. We are thus led to formulate ten conjectures, each of which can be viewed as a multivariate quadratic transformation, and can be proved in a number of special cases.

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Additional Information:© 2012 Elsevier Inc. Received 1 June 2011. Available online 23 March 2012. The author would like to thank V. Spiridonov and O. Warnaar for helpful comments on an earlier draft, along with R. Askey and M. Rahman for helpful discussions regarding the analogue of Corollary 4.12 for Askey–Wilson polynomials, and F. van de Bult for settling one conjecture from the original version and providing substantial additional evidence for several more. The author would also like to thank P. Forrester for hosting the author’s sabbatical at the University of Melbourne, during which several key portions of the work were done. This work was supported in part by NSF Grants Nos. DMS-0401387, DMS-0833464, and DMS-1001645; in addition, much of the work was performed while the author was affiliated with the University of California at Davis.
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Subject Keywords:Elliptic special functions; Macdonald polynomials; Koornwinder polynomials; Quadratic transformations
Issue or Number:7
Record Number:CaltechAUTHORS:20120730-134423880
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Official Citation:Eric M. Rains, Elliptic Littlewood identities, Journal of Combinatorial Theory, Series A, Volume 119, Issue 7, October 2012, Pages 1558-1609, ISSN 0097-3165, 10.1016/j.jcta.2012.03.001.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:32794
Deposited By: Tony Diaz
Deposited On:31 Jul 2012 21:22
Last Modified:09 Nov 2021 21:30

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