On the Faraut-Koranyi hypergeometric functions in rank two
[Sur les fonctions hypergéométriques de Faraut-Koranyi en rang deux]
Annales de l'Institut Fourier, Tome 54 (2004) no. 6, pp. 1855-1875.

Nous donnons une description complète du comportement à la frontière des fonctions hypergéométriques généralisées introduites par Faraut et Koranyi sur les domaines de Cartan de rang deux. Le principal outil est une nouvelle représentation intégrale pour certains polynômes sphériques, qui peut avoir un intérêt dans d'autres contextes.

We give a complete description of the boundary behaviour of the generalized hypergeometric functions, introduced by Faraut and Koranyi, on Cartan domains of rank 2. The main tool is a new integral representation for some spherical polynomials, which may be of independent interest.

DOI : https://doi.org/10.5802/aif.2069
Classification : 33D67,  32M15,  33C67
Mots clés : domaine de Cartan, fonction hypergéométrique, partition, polynôme sphérique, polynôme de Jack
@article{AIF_2004__54_6_1855_0,
     author = {Engli\v{s}, Miroslav and Zhang, Genkai},
     title = {On the Faraut-Koranyi hypergeometric functions in rank two},
     journal = {Annales de l'Institut Fourier},
     pages = {1855--1875},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {6},
     year = {2004},
     doi = {10.5802/aif.2069},
     zbl = {1079.33010},
     mrnumber = {2134227},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2069/}
}
Engliš, Miroslav; Zhang, Genkai. On the Faraut-Koranyi hypergeometric functions in rank two. Annales de l'Institut Fourier, Tome 54 (2004) no. 6, pp. 1855-1875. doi : 10.5802/aif.2069. http://www.numdam.org/articles/10.5802/aif.2069/

[Ar] J. Arazy; R.E. Curto, R.G. Douglas, J.D. Pincus, N. Salinas A survey of invariant Hilbert spaces of analytic functions on bounded symmetric domains, Multivariable operator theory (Contemporary Mathematics), Volume vol. 185 (1995), pp. 7-65 | Zbl 0831.46014

[BE] H. Bateman; A. Erdélyi Higher transcendental functions, vol. I, McGraw-Hill, New York -- Toronto -- London, 1953 | MR 58756

[CR] R.R. Coifman; R. Rochberg Representation theorems for Hardy spaces, Asterisque, Volume 77 (1980), pp. 11-66 | MR 604369 | Zbl 0472.46040

[E] M. Engliš Compact Toeplitz operators via the Berezin transform on bounded symmetric domains, Integral Eq. Oper. Theory, Volume 33 (1999), pp. 426-455 | MR 1682815 | Zbl 0936.47014

[E] M. Engliš Compact Toeplitz operators via the Berezin transform on bounded symmetric domains, Integral Eq. Oper. Theory (Erratum Ibid), Volume 34 (1999), p. 500-501 | MR 1702236 | Zbl 0936.47014

[FK] J. Faraut; A. Koranyi Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal, Volume 88 (1990), pp. 64-89 | MR 1033914 | Zbl 0718.32026

[H] S. Helgason Groups and geometric analysis, Academic Press, Orlando, 1984 | MR 754767 | Zbl 0543.58001

[Lo] O. Loos Bounded symmetric domains and Jordan pairs, Irvine, University of California, 1977

[MD] I.G. MacDonald Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1995 | MR 1354144 | Zbl 0824.05059

[Sa] P. Sawyer Spherical functions on symmetric cones, Trans. Amer. Math. Soc, Volume 349 (1997), pp. 3569-3584 | MR 1325919 | Zbl 0881.33011

[Sh] N. Shimeno Boundary value problems for the Shilov boundary of a bounded symmetric domain of tube type, J. Funct. Anal, Volume 140 (1996), pp. 124-141 | MR 1404577 | Zbl 0857.43008

[Up] H. Upmeier Toeplitz operators on bounded symmetric domains, Trans. Amer. Math. Soc, Volume 280 (1983), pp. 221-237 | MR 712257 | Zbl 0527.47019

[Y] Z. Yan A class of generalized hypergeometric functions in several variables, Canad. J. Math, Volume 44 (1992), pp. 1317-1338 | MR 1192421 | Zbl 0769.33014

[Zh] K. Zhu Holomorphic Besov spaces on bounded symmetric domains, Quart. J. Math. Oxford, Volume 46 (1995), pp. 239-256 | MR 1333834 | Zbl 0837.32013