146 research outputs found
Large Parameter Cases of the Gauss Hypergeometric Function
We consider the asymptotic behaviour of the Gauss hypergeometric function
when several of the parameters a, b, c are large. We indicate which cases are
of interest for orthogonal polynomials (Jacobi, but also Krawtchouk, Meixner,
etc.), which results are already available and which cases need more attention.
We also consider a few examples of 3F2-functions of unit argument, to explain
which difficulties arise in these cases, when standard integrals or
differential equations are not available.Comment: 21 pages, 4 figure
On generalized Cauchy-Stieltjes transforms of some Beta distributions
We express generalized Cauchy-Stieltjes transforms of some particular Beta
distributions (of ultraspherical type generating functions for orthogonal
polynomials) as a powered Cauchy-Stieltjes transform of some measure. For
suitable values of the power parameter, the latter measure turns out to be a
probability measure and its density is written down using Markov transforms.
The discarded values give a negative answer to a deformed free probability
unless a restriction on the power parameter is made. A particular symmetric
distribution interpolating between Wigner and arcsine distributions is
obtained. Its moments are expressed through a terminating hypergeometric series
interpolating between Catalan and shifed Catalan numbers. for small values of
the power parameter, the free cumulants are computed. Interesting opne problems
related to a deformed representation theory of the infinite symmetric group and
to a deformed Bozejko's convolution are discussed
The Wigner function of a q-deformed harmonic oscillator model
The phase space representation for a q-deformed model of the quantum harmonic
oscillator is constructed. We have found explicit expressions for both the
Wigner and Husimi distribution functions for the stationary states of the
-oscillator model under consideration. The Wigner function is expressed as a
basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is
shown that, in the limit case (), both the Wigner and Husimi
distribution functions reduce correctly to their well-known non-relativistic
analogues. Surprisingly, examination of both distribution functions in the
q-deformed model shows that, when , their behaviour in the phase space
is similar to the ground state of the ordinary quantum oscillator, but with a
displacement towards negative values of the momentum. We have also computed the
mean values of the position and momentum using the Wigner function. Unlike the
ordinary case, the mean value of the momentum is not zero and it depends on
and . The ground-state like behaviour of the distribution functions for
excited states in the q-deformed model opens quite new perspectives for further
experimental measurements of quantum systems in the phase space.Comment: 16 pages, 24 EPS figures, uses IOP style LaTeX, some misprints are
correctd and journal-reference is adde
A Probablistic Origin for a New Class of Bivariate Polynomials
We present here a probabilistic approach to the generation of new polynomials
in two discrete variables. This extends our earlier work on the 'classical'
orthogonal polynomials in a previously unexplored direction, resulting in the
discovery of an exactly soluble eigenvalue problem corresponding to a bivariate
Markov chain with a transition kernel formed by a convolution of simple
binomial and trinomial distributions. The solution of the relevant
eigenfunction problem, giving the spectral resolution of the kernel, leads to
what we believe to be a new class of orthogonal polynomials in two discrete
variables. Possibilities for the extension of this approach are discussed.Comment: This is a contribution to the Special Issue on Dunkl Operators and
Related Topics, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
- …