, Volume 7, Issue 5, pp 1569-1581
Date: 08 Nov 2012

True Poly-Bergman and Poly-Bergman Kernels for the Complement of a Closed Disk

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Abstract

Let \(k\) and \(j\) be positive integers. We prove that the action of the two-dimensional singular integral operators \((S_\Omega )^{j-1}\) and \((S_\Omega ^*)^{j-1}\) on a Hilbert base for the Bergman space \(\mathcal{A }^2(\Omega )\) and anti-Bergman space \(\mathcal{A }^2_{-1}(\Omega ),\) respectively, gives Hilbert bases \(\{ \psi _{\pm j , k } \}_{ k }\) for the true poly-Bergman spaces \(\mathcal{A }_{(\pm j)}^2(\Omega ),\) where \(S_\Omega \) denotes the compression of the Beurling transform to the Lebesgue space \(L^2(\Omega , dA).\) The functions \(\psi _{\pm j,k}\) will be explicitly represented in terms of the \((2,1)\) -hypergeometric polynomials as well as by formulas of Rodrigues type. We prove explicit representations for the true poly-Bergman kernels and more transparent representations for the poly-Bergman kernels of \(\Omega \) . We establish Rodrigues type formulas for the poly-Bergman kernels of \(\mathbb{D }\) .

Communicated by Laurent Baratchart.