187 research outputs found
Matrix-Valued Little q-Jacobi Polynomials
Matrix-valued analogues of the little q-Jacobi polynomials are introduced and
studied. For the 2x2-matrix-valued little q-Jacobi polynomials explicit
expressions for the orthogonality relations, Rodrigues formula, three-term
recurrence relation and their relation to matrix-valued q-hypergeometric series
and the scalar-valued little q-Jacobi polynomials are presented. The study is
based on a matrix-valued q-difference operator, which is a q-analogue of
Tirao's matrix-valued hypergeometric differential operator.Comment: 16 pages, various corrections and minor additions, incorporating
referee's comment
Noncommutative Gauge Theory on the q-Deformed Euclidean Plane
In this talk we recall some concepts of Noncommutative Gauge Theories. In
particular, we discuss the q-deformed two-dimensional Euclidean Plane which is
covariant with respect to the q-deformed Euclidean group. A Seiberg-Witten map
is constructed to express noncommutative fields in terms of their commutative
counterparts.Comment: 5 pages; Talk given by Frank Meyer at the 9th Adriatic Meeting,
September 4th-14th, 2003, Dubrovni
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
Properties of generalized univariate hypergeometric functions
Based on Spiridonov's analysis of elliptic generalizations of the Gauss
hypergeometric function, we develop a common framework for 7-parameter families
of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric
functions. In each case we derive the symmetries of the generalized
hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic)
and of type E_6 (trigonometric) using the appropriate versions of the
Nassrallah-Rahman beta integral, and we derive contiguous relations using
fundamental addition formulas for theta and sine functions. The top level
degenerations of the hyperbolic and trigonometric hypergeometric functions are
identified with Ruijsenaars' relativistic hypergeometric function and the
Askey-Wilson function, respectively. We show that the degeneration process
yields various new and known identities for hyperbolic and trigonometric
special functions. We also describe an intimate connection between the
hyperbolic and trigonometric theory, which yields an expression of the
hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric
hypergeometric functions.Comment: 46 page
Big q-Laguerre and q-Meixner polynomials and representations of the algebra U_q(su(1,1))
Diagonalization of a certain operator in irreducible representations of the
positive discrete series of the quantum algebra U_q(su(1,1)) is studied.
Spectrum and eigenfunctions of this operator are found in an explicit form.
These eigenfunctions, when normalized, constitute an orthonormal basis in the
representation space. The initial U_q(su(1,1))-basis and the basis of
eigenfunctions are interrelated by a matrix with entries, expressed in terms of
big q-Laguerre polynomials. The unitarity of this connection matrix leads to an
orthogonal system of functions, which are dual with respect to big q-Laguerre
polynomials. This system of functions consists of two separate sets of
functions, which can be expressed in terms of q-Meixner polynomials
M_n(x;b,c;q) either with positive or negative values of the parameter b. The
orthogonality property of these two sets of functions follows directly from the
unitarity of the connection matrix. As a consequence, one obtains an
orthogonality relation for q-Meixner polynomials M_n(x;b,c;q) with b<0. A
biorthogonal system of functions (with respect to the scalar product in the
representation space) is also derived.Comment: 15 pages, LaTe
The oscillator model for the Lie superalgebra sh(2|2) and Charlier polynomials
We investigate an algebraic model for the quantum oscillator based upon the
Lie superalgebra sh(2|2), known as the Heisenberg-Weyl superalgebra or "the
algebra of supersymmetric quantum mechanics", and its Fock representation. The
model offers some freedom in the choice of a position and a momentum operator,
leading to a free model parameter gamma. Using the technique of Jacobi
matrices, we determine the spectrum of the position operator, and show that its
wavefunctions are related to Charlier polynomials C_n with parameter gamma^2.
Some properties of these wavefunctions are discussed, as well as some other
properties of the current oscillator model.Comment: Minor changes and some additional references added in version
Quantum Field Theory on the Noncommutative Plane with Symmetry
We study properties of a scalar quantum field theory on the two-dimensional
noncommutative plane with quantum symmetry. We start from the
consideration of a firstly quantized quantum particle on the noncommutative
plane. Then we define quantum fields depending on noncommutative coordinates
and construct a field theoretical action using the -invariant measure
on the noncommutative plane. With the help of the partial wave decomposition we
show that this quantum field theory can be considered as a second quantization
of the particle theory on the noncommutative plane and that this field theory
has (contrary to the common belief) even more severe ultraviolet divergences
than its counterpart on the usual commutative plane. Finally, we introduce the
symmetry transformations of physical states on noncommutative spaces and
discuss them in detail for the case of the quantum group.Comment: LaTeX, 26 page
Free Meixner states
Free Meixner states are a class of functionals on non-commutative polynomials
introduced in math.CO/0410482. They are characterized by a resolvent-type form
for the generating function of their orthogonal polynomials, by a recursion
relation for those polynomials, or by a second-order non-commutative
differential equation satisfied by their free cumulant functional. In this
paper, we construct an operator model for free Meixner states. By combinatorial
methods, we also derive an operator model for their free cumulant functionals.
This, in turn, allows us to construct a number of examples. Many of these
examples are shown to be trivial, in the sense of being free products of
functionals which depend on only a single variable, or rotations of such free
products. On the other hand, the multinomial distribution is a free Meixner
state and is not a product. Neither is a large class of tracial free Meixner
states which are analogous to the simple quadratic exponential families in
statistics.Comment: 30 page
LU factorizations, q=0 limits, and p-adic interpretations of some q-hypergeometric orthogonal polynomials
For little q-Jacobi polynomials and q-Hahn polynomials we give particular
q-hypergeometric series representations in which the termwise q=0 limit can be
taken. When rewritten in matrix form, these series representations can be
viewed as LU factorizations. We develop a general theory of LU factorizations
related to complete systems of orthogonal polynomials with discrete
orthogonality relations which admit a dual system of orthogonal polynomials.
For the q=0 orthogonal limit functions we discuss interpretations on p-adic
spaces. In the little 0-Jacobi case we also discuss product formulas.Comment: changed title, references updated, minor changes matching the version
to appear in Ramanujan J.; 22 p
Spectral properties of operators using tridiagonalisation
A general scheme for tridiagonalising differential, difference or
q-difference operators using orthogonal polynomials is described. From the
tridiagonal form the spectral decomposition can be described in terms of the
orthogonality measure of generally different orthogonal polynomials. Three
examples are worked out: (1) related to Jacobi and Wilson polynomials for a
second order differential operator, (2) related to little q-Jacobi polynomials
and Askey-Wilson polynomials for a bounded second order q-difference operator,
(3) related to little q-Jacobi polynomials for an unbounded second order
q-difference operator. In case (1) a link with the Jacobi function transform is
established, for which we give a q-analogue using example (2).Comment: 14 pages, corrections, to appear in Analysis and Application
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