6,122 research outputs found
Berezin transform on the quantum unit ball
We introduce and study, in the framework of a theory of quantum Cartan
domains, a q-analogue of the Berezin transform on the unit ball. We construct
q-analogues of weighted Bergman spaces, Toeplitz operators and covariant symbol
calculus. In studying the analytical properties of the Berezin transform we
introduce also the q-analogue of the SU(n,1)-invariant Laplace operator (the
Laplace-Beltrami operator) and present related results on harmonic analysis on
the quantum ball. These are applied to obtain an analogue of one result by
A.Unterberger and H.Upmeier. An explicit asymptotic formula expressing the
q-Berezin transform via the q-Laplace-Beltrami operator is also derived. At the
end of the paper, we give an application of our results to basic hypergeometric
q-orthogonal polynomials.Comment: 38 pages, accepted by Journal of Mathematical Physic
Elliptic U(2) quantum group and elliptic hypergeometric series
We investigate an elliptic quantum group introduced by Felder and Varchenko,
which is constructed from the -matrix of the Andrews-Baxter-Forrester model,
containing both spectral and dynamical parameter. We explicitly compute the
matrix elements of certain corepresentations and obtain orthogonality relations
for these elements. Using dynamical representations these orthogonality
relations give discrete bi-orthogonality relations for terminating
very-well-poised balanced elliptic hypergeometric series, previously obtained
by Frenkel and Turaev and by Spiridonov and Zhedanov in different contexts.Comment: 20 page
On inversion and connection coefficients for basic hypergeometric polynomials
In this paper, we propose a general method to express explicitly the
inversion and the connection coefficients between two basic hypergeometric
polynomial sets. As application, we consider some -orthogonal basic
hypergeometric polynomials and we derive expansion formulae corresponding to
all the families within the -Askey scheme.Comment: 15 page
Wilson function transforms related to Racah coefficients
The irreducible -representations of the Lie algebra consist of
discrete series representations, principal unitary series and complementary
series. We calculate Racah coefficients for tensor product representations that
consist of at least two discrete series representations. We use the explicit
expressions for the Clebsch-Gordan coefficients as hypergeometric functions to
find explicit expressions for the Racah coefficients. The Racah coefficients
are Wilson polynomials and Wilson functions. This leads to natural
interpretations of the Wilson function transforms. As an application several
sum and integral identities are obtained involving Wilson polynomials and
Wilson functions. We also compute Racah coefficients for U_q(\su(1,1)), which
turn out to be Askey-Wilson functions and Askey-Wilson polynomials.Comment: 48 page
Harmonic analysis on the SU(2) dynamical quantum group
Dynamical quantum groups were recently introduced by Etingof and Varchenko as
an algebraic framework for studying the dynamical Yang-Baxter equation, which
is precisely the Yang-Baxter equation satisfied by 6j-symbols. We investigate
one of the simplest examples, generalizing the standard SU(2) quantum group.
The matrix elements for its corepresentations are identified with Askey-Wilson
polynomials, and the Haar measure with the Askey-Wilson measure. The discrete
orthogonality of the matrix elements yield the orthogonality of q-Racah
polynomials (or quantum 6j-symbols). The Clebsch-Gordan coefficients for
representations and corepresentations are also identified with q-Racah
polynomials. This results in new algebraic proofs of the Biedenharn-Elliott
identity satisfied by quantum 6j-symbols.Comment: 51 pages; minor correction
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