On Function Theory in Quantum Disc: Integral Representations

L. L. Vaksman, D. L. Shklyarov, Sergey D. Sinelshchikov

Abstract: The present work considers one of the simplest homogeneous spaces of the quantum group SU(1,1), the q-analogue of the unit disc in ${Bbb C}$. We state without proofs q-analogues of Cauchy-Green formulae, integral representations of eigenfunctions of the Laplace-Beltrami operator, Green functions for Poisson equation and an inversion formula for Fourier transform. It is also demonstrated that the two-parameter quantization of the disc introduced before by S. Klimec and A. Lesniewski, can be derived via an application of the method of F. Berezin.

Mathematics Subject Classification ID

Quantum groups and related algebraic methods applied to problems in quantum theory (81R50) General mathematical topics and methods in quantum theory (81Q99)

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