Generalized q-Hermite Polynomials |
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Authors: | Christian Berg Andreas Ruffing |
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Affiliation: | Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen,?Denmark. E-mail: berg@math.ku.dk, DK Zentrum Mathematik, Technische Universit?t München, Arcisstrasse 21, 80333 München, Germany.?E-mail: ruffing@appl-math.tu-muenchen.de, DE
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Abstract: | We consider two operators A and A + in a Hilbert space of functions on the exponential lattice , where 0<q<1. The operators are formal adjoints of each other and depend on a real parameter . We show how these operators lead to an essentially unique symmetric ground state ψ0 and that A and A + are ladder operators for the sequence . The sequence (ψ n /ψ0) is shown to be a family of orthogonal polynomials, which we identify as symmetrized q-Laguerre polynomials. We obtain in this way a new proof of the orthogonality for these polynomials. When γ=0 the polynomials are the discrete q-Hermite polynomials of type II, studied in several papers on q-quantum mechanics. Received: 6 December 1999 / Accepted: 21 May 2001 |
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