CENTER FOR APPLIED MATHEMATICS AND THEORETICAL PHYSICS
UNIVERZA V MARIBORU UNIVERSITY OF MARIBOR
KREKOVA 2 SI-2000 MARIBOR SLOVENIA
Phone +(386) (2) 2355 350 and 2355 351 $\bullet$ Fax +(386) (2) 2355 360
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PROF.DR. MARKO ROBNIK, DIRECTOR

Seminarsko predavanje
Centra za uporabno matematiko in teoreticno fiziko

Generalized $q$-Hermite Polynomials

Dr. Andreas Ruffing

Technische Universität München, Zentrum Mathematik
Arcisstrasse 21, D-80333 München, Germany

Abstract In this talk we refer to the results of an article by Christian Berg and Andreas Ruffing which will be published in Commun. Math. Phys. We consider two operators $A$ and $A^+$ in a Hilbert space of functions on the exponential lattice $\{q^n,-q^n\vert n\in \Bbb Z\}$ where $0<q<1$. The operators are formal adjoints of each other and depend on a real parameter $\gamma<\frac12$. We show how these operators lead to an essentially unique symmetric ground state $\psi_0$ and that $A$ and $A^+$ are ladder operators for the sequence $\psi_n=(A^+)^n\psi_0$. The sequence $(\psi_n/\psi_0)$ 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 $\gamma=0$ the polynomials are the discrete $q$-Hermite polynomials of type II, studied in several papers on $q$-quantum mechanics.

Seminarsko predavanje bo v sredo, 22. avgusta 2001 ob 15:15 uri v se- minarski sobi CAMTP, Krekova 2, pritlicje. Vljudno vabljeni vsi zainteresirani, tudi študentje.

Prof.Dr. Marko Robnik
-- Direktor CAMTP --