On the Faraut-Koranyi hypergeometric functions in rank two
Annales de l'Institut Fourier, Volume 54 (2004) no. 6, p. 1855-1875
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.
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.
DOI : https://doi.org/10.5802/aif.2069
Classification:  33D67,  32M15,  33C67
Keywords: Cartan domain, hypergeometric function, partition, spherical polynomial, Jack polynomial
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {6},
     year = {2004},
     pages = {1855-1875},
     doi = {10.5802/aif.2069},
     zbl = {1079.33010},
     mrnumber = {2134227},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2004__54_6_1855_0}
}
Engliš, Miroslav; Zhang, Genkai. On the Faraut-Koranyi hypergeometric functions in rank two. Annales de l'Institut Fourier, Volume 54 (2004) no. 6, pp. 1855-1875. doi : 10.5802/aif.2069. http://www.numdam.org/item/AIF_2004__54_6_1855_0/

[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, Amer. Math. Soc., Providence (Contemporary Mathematics) Tome vol. 185 (1995), pp. 7-65 | Zbl 0831.46014

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

[CR] R.R. Coifman; R. Rochberg Representation theorems for Hardy spaces, Asterisque, Tome 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, Tome 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), Tome 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, Tome 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, Tome 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, Tome 140 (1996), pp. 124-141 | MR 1404577 | Zbl 0857.43008

[Up] H. Upmeier Toeplitz operators on bounded symmetric domains, Trans. Amer. Math. Soc, Tome 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, Tome 44 (1992), pp. 1317-1338 | MR 1192421 | Zbl 0769.33014

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