# Limit transition between hypergeometric functions of type BC and type A

M. Rösler, T. Koornwinder, M. Voit, Compositio Mathematica 149 (2013) 1381–1400.

Journal Article | Published | English
Author
Rösler, MargitLibreCat; Koornwinder, Tom; Voit, Michael
Department
Abstract
<jats:title>Abstract</jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007045_inline1" /><jats:tex-math>${F}_{BC} (\lambda , k; t)$</jats:tex-math></jats:alternatives></jats:inline-formula> be the Heckman–Opdam hypergeometric function of type BC with multiplicities <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007045_inline2" /><jats:tex-math>$k= ({k}_{1} , {k}_{2} , {k}_{3} )$</jats:tex-math></jats:alternatives></jats:inline-formula> and weighted half-sum <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007045_inline3" /><jats:tex-math>$\rho (k)$</jats:tex-math></jats:alternatives></jats:inline-formula> of positive roots. We prove that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007045_inline4" /><jats:tex-math>${F}_{BC} (\lambda + \rho (k), k; t)$</jats:tex-math></jats:alternatives></jats:inline-formula> converges as <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007045_inline5" /><jats:tex-math>${k}_{1} + {k}_{2} \rightarrow \infty$</jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007045_inline6" /><jats:tex-math>${k}_{1} / {k}_{2} \rightarrow \infty$</jats:tex-math></jats:alternatives></jats:inline-formula> to a function of type A for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007045_inline7" /><jats:tex-math>$t\in { \mathbb{R} }^{n}$</jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007045_inline8" /><jats:tex-math>$\lambda \in { \mathbb{C} }^{n}$</jats:tex-math></jats:alternatives></jats:inline-formula>. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007045_inline9" /><jats:tex-math>$\mathbb{F} = \mathbb{R} , \mathbb{C} , \mathbb{H}$</jats:tex-math></jats:alternatives></jats:inline-formula> when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski.</jats:p>
Keywords
Publishing Year
Journal Title
Compositio Mathematica
Volume
149
Issue
8
Page
1381-1400
ISSN
LibreCat-ID

### Cite this

Rösler M, Koornwinder T, Voit M. Limit transition between hypergeometric functions of type BC and type A. Compositio Mathematica. 2013;149(8):1381-1400. doi:10.1112/s0010437x13007045
Rösler, M., Koornwinder, T., & Voit, M. (2013). Limit transition between hypergeometric functions of type BC and type A. Compositio Mathematica, 149(8), 1381–1400. https://doi.org/10.1112/s0010437x13007045
@article{Rösler_Koornwinder_Voit_2013, title={Limit transition between hypergeometric functions of type BC and type A}, volume={149}, DOI={10.1112/s0010437x13007045}, number={8}, journal={Compositio Mathematica}, publisher={Wiley}, author={Rösler, Margit and Koornwinder, Tom and Voit, Michael}, year={2013}, pages={1381–1400} }
Rösler, Margit, Tom Koornwinder, and Michael Voit. “Limit Transition between Hypergeometric Functions of Type BC and Type A.” Compositio Mathematica 149, no. 8 (2013): 1381–1400. https://doi.org/10.1112/s0010437x13007045.
M. Rösler, T. Koornwinder, and M. Voit, “Limit transition between hypergeometric functions of type BC and type A,” Compositio Mathematica, vol. 149, no. 8, pp. 1381–1400, 2013, doi: 10.1112/s0010437x13007045.
Rösler, Margit, et al. “Limit Transition between Hypergeometric Functions of Type BC and Type A.” Compositio Mathematica, vol. 149, no. 8, Wiley, 2013, pp. 1381–400, doi:10.1112/s0010437x13007045.

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