Spherical harmonic expansion of the Poisson-Szegő kernel for the ball

Folland, G. B.

doi:10.1090/S0002-9939-1975-0370044-2

Spherical harmonic expansion of the Poisson-Szegő kernel for the ball
HTML articles powered by AMS MathViewer

by G. B. Folland PDF

Proc. Amer. Math. Soc. 47 (1975), 401-408 Request permission

Abstract:

By applying the theory of unitary spherical harmonics, we obtain an expansion of the Poisson-Szegö kernel for the unit ball in complex $n$-space in terms of Jacobi polynomials and hypergeometric functions.

References

Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
G. B. Folland, The tangential Cauchy-Riemann complex on spheres, Trans. Amer. Math. Soc. 171 (1972), 83–133. MR 309156, DOI 10.1090/S0002-9947-1972-0309156-X
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by Leo Ebner and Adam Korányi. MR 0171936, DOI 10.1090/mmono/006
Mineo Ikeda, On spherical functions for the unitary group. I. General theory, Mem. Fac. Engrg. Hiroshima Univ. 3 (1967), no. 1, 17–29 (1967). MR 220990

The addition formula for Jacobi polynomials

E. M. Stein, Boundary behavior of holomorphic functions of several complex variables, Mathematical Notes, No. 11, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0473215
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
N. Ja. Vilenkin and R. L. Šapiro, Irreducible representations of the group $\textrm {SU}(n)$ of class I relative to $\textrm {SU}(n-1)$, Izv. Vysš. Učebn. Zaved. Matematika 1967 (1967), no. 7 (62), 9–20 (Russian). MR 0219662