Polynomials of Meixner's type in infinite dimensions—Jacobi fields and orthogonality measures |
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Authors: | Eugene Lytvynov |
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Institution: | a Institut für Angewandte Mathematik, Abteilung fur Stochastik, Universität Bonn, Wegelerstrasse 6, D-53115 Bonn, Germany b SFB 611, Universität Bonn, Germany c BiBoS, Universität Bielefeld, Germany |
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Abstract: | The classical polynomials of Meixner's type—Hermite, Charlier, Laguerre, Meixner, and Meixner-Pollaczek polynomials—are distinguished through a special form of their generating function, which involves the Laplace transform of their orthogonality measure. In this paper, we study analogs of the latter three classes of polynomials in infinite dimensions. We fix as an underlying space a (non-compact) Riemannian manifold X and an intensity measure σ on it. We consider a Jacobi field in the extended Fock space over L2(X;σ), whose field operator at a point x∈X is of the form , where λ is a real parameter. Here, ∂x and are, respectively, the annihilation and creation operators at the point x. We then realize the field operators as multiplication operators in , where is the dual of , and μλ is the spectral measure of the Jacobi field. We show that μλ is a gamma measure for |λ|=2, a Pascal measure for |λ|>2, and a Meixner measure for |λ|<2. In all the cases, μλ is a Lévy noise measure. The isomorphism between the extended Fock space and is carried out by infinite-dimensional polynomials of Meixner's type. We find the generating function of these polynomials and using it, we study the action of the operators ∂x and in the functional realization. |
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Keywords: | primary 60G51 60G57 secondary 60H40 47B36 |
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