Hermitian symmetric spaces of tube type and multivariate Meixner-Pollaczek polynomials


  • Jacques Faraut
  • Masato Wakayama




Harmonic analysis on Hermitian symmetric spaces of tube type is a natural framework for introducing multivariate Meixner-Pollaczek polynomials. Their main properties are established in this setting: orthogonality, generating and determinantal formulae, difference equations. For proving these properties we use the composition of the following transformations: Cayley transform, Laplace transform, and spherical Fourier transform associated to Hermitian symmetric spaces of tube type.  In particular the difference equation for the multivariate Meixner-Pollaczek polynomials is obtained from an Euler type equation on a bounded symmetric domain.


Andrews, G. E., Askey, R., and Roy, R., Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. http://dx.doi.org/10.1017/CBO9781107325937

Aristidou, M., Davidson, M., and Ólafsson, G., Laguerre functions on symmetric cones and recursion relations in the real case, J. Comput. Appl. Math. 199 (2007), no. 1, 95–112. http://dx.doi.org/10.1016/j.cam.2005.12.002

Baker, T. H. and Forrester, P. J., The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), no. 1, 175–216. http://dx.doi.org/10.1007/s002200050161

Davidson, M. and Ólafsson, G., Differential recursion relations for Laguerre functions on Hermitian matrices, Integral Transforms Spec. Funct. 14 (2003), no. 6, 469–484. http://dx.doi.org/10.1080/10652460310001600582

Davidson, M., Ólafsson, G., and Zhang, G., Laplace and Segal-Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials, J. Funct. Anal. 204 (2003), no. 1, 157–195. http://dx.doi.org/10.1016/S0022-1236(03)00101-0

Dib, H., Fonctions de Bessel sur une algèbre de Jordan, J. Math. Pures Appl. (9) 69 (1990), no. 4, 403–448.

Faraut, J., Analysis on the crown of a Riemannian symmetric space, Lie groups and symmetric spaces, Amer. Math. Soc. Transl. Ser. 2, vol. 210, Amer. Math. Soc., Providence, RI, 2003, pp. 99--110. http://dx.doi.org/10.1090/trans2/210/07

Faraut, J. and Korányi, A., Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, 1994.

Hua, L. K., Harmonic analysis of functions of several complex variables in the classical domains, American Mathematical Society, Providence, R.I., 1963.

Lassalle, M., Coefficients binomiaux généralisés et polynômes de Macdonald, J. Funct. Anal. 158 (1998), no. 2, 289–324. http://dx.doi.org/10.1006/jfan.1998.3281

Ø rsted, B. and Zhang, G. K., Weyl quantization and tensor products of Fock and Bergman spaces, Indiana Univ. Math. J. 43 (1994), no. 2, 551–583. http://dx.doi.org/10.1512/iumj.1994.43.43023

Peetre, J. and Zhang, G. K., A weighted Plancherel formula. III. The case of the hyperbolic matrix ball, Collect. Math. 43 (1992), no. 3, 273–301 (1993).

Sahi, S. and Zhang, G., Biorthogonal expansion of non-symmetric Jack functions, SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 106, 9. http://dx.doi.org/10.3842/SIGMA.2007.106

Schoutens, W., Stochastic processes and orthogonal polynomials, Lecture Notes in Statistics, vol. 146, Springer-Verlag, New York, 2000. http://dx.doi.org/10.1007/978-1-4612-1170-9

Zhang, G., Invariant differential operators on symmetric cones and Hermitian symmetric spaces, Acta Appl. Math. 73 (2002), no. 1-2, 79–94. http://dx.doi.org/10.1023/A:1019722703356

Zhang, G. K., Some recurrence formulas for spherical polynomials on tube domains, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1725–1734. http://dx.doi.org/10.2307/2154967




How to Cite

Faraut, J., & Wakayama, M. (2017). Hermitian symmetric spaces of tube type and multivariate Meixner-Pollaczek polynomials. MATHEMATICA SCANDINAVICA, 120(1), 87–114. https://doi.org/10.7146/math.scand.a-25506