39 research outputs found

### New connection formulae for some q-orthogonal polynomials in q-Askey scheme

New nonlinear connection formulae of the q-orthogonal polynomials, such
continuous q-Laguerre, continuous big q-Hermite, q-Meixner-Pollaczek and
q-Gegenbauer polynomials, in terms of their respective classical analogues are
obtained using a special realization of the q-exponential function as infinite
multiplicative series of ordinary exponential function

### Wavelets in mathematical physics: q-oscillators

We construct representations of a q-oscillator algebra by operators on Fock
space on positive matrices. They emerge from a multiresolution scaling
construction used in wavelet analysis. The representations of the Cuntz Algebra
arising from this multiresolution analysis are contained as a special case in
the Fock Space construction.Comment: (03/11/03):18 pages; LaTeX2e, "article" document class with
"letterpaper" option An outline was added under the abstract (p.1),
paragraphs added to Introduction (p.2), mat'l added to Proofs in Theorems 1
and 6 (pgs.5&17), material added to text for the conclusion (p.17), one add'l
reference added [12]. (04/22/03):"number 1" replace with "term C" (p.9),
single sentences reformed into a one paragraph (p.13), QED symbol moved up
one paragraph and last paragraph labeled as "Concluding Remarks.

### Interpolation of SUSY quantum mechanics

Interpolation of two adjacent Hamiltonians in SUSY quantum mechanics
$H_s=(1-s)A^{\dagger}A + sAA^{\dagger}$, $0\le s\le 1$ is discussed together
with related operators. For a wide variety of shape-invariant degree one
quantum mechanics and their `discrete' counterparts, the interpolation
Hamiltonian is also shape-invariant, that is it takes the same form as the
original Hamiltonian with shifted coupling constant(s).Comment: 18 page

### Quantum state transfer in spin chains with q-deformed interaction terms

We study the time evolution of a single spin excitation state in certain
linear spin chains, as a model for quantum communication. Some years ago it was
discovered that when the spin chain data (the nearest neighbour interaction
strengths and the magnetic field strengths) are related to the Jacobi matrix
entries of Krawtchouk polynomials or dual Hahn polynomials, so-called perfect
state transfer takes place. The extension of these ideas to other types of
discrete orthogonal polynomials did not lead to new models with perfect state
transfer, but did allow more insight in the general computation of the
correlation function. In the present paper, we extend the study to discrete
orthogonal polynomials of q-hypergeometric type. A remarkable result is a new
analytic model where perfect state transfer is achieved: this is when the spin
chain data are related to the Jacobi matrix of q-Krawtchouk polynomials. The
other cases studied here (affine q-Krawtchouk polynomials, quantum q-Krawtchouk
polynomials, dual q-Krawtchouk polynomials, q-Hahn polynomials, dual q-Hahn
polynomials and q-Racah polynomials) do not give rise to models with perfect
state transfer. However, the computation of the correlation function itself is
quite interesting, leading to advanced q-series manipulations

### Action of overalgebra in Plancherel decomposition and shift operators in imaginary direction

Consider the Plancherel decomposition of the tensor product of a highest
weight and a lowest weight unitary representations of $SL_2$.
We construct explicitly the action of the Lie algebra $sl_2 + sl_2$ in the
direct integral of Hilbert spaces. It turns out that a Lie algebra operator is
a second order differential operator in one variable and second order
difference operator with respect to another variable. The difference operators
are defined in terms of the shift in the imaginary direction $f(s)\mapsto
f(s+i)$, $i^2=-1$ (the Plancherel measure is supported by real $s$).Comment: 12 page

### On factorization of q-difference equation for continuous q-Hermite polynomials

We argue that a customary q-difference equation for the continuous q-Hermite
polynomials H_n(x|q) can be written in the factorized form as (D_q^2 -
1)H_n(x|q)=(q^{-n}-1)H_n(x|q), where D_q is some explicitly known q-difference
operator. This means that the polynomials H_n(x|q) are in fact governed by the
q-difference equation D_qH_n(x|q)=q^{-n/2}H_n(x|q), which is simpler than the
conventional one.Comment: 7 page

### Green function on the quantum plane

Green function (which can be called the q-analogous of the Hankel function)
on the quantum plane E_q^2= E_q(2)/U(1) is constructed.Comment: 8 page

### A "missing" family of classical orthogonal polynomials

We study a family of "classical" orthogonal polynomials which satisfy (apart
from a 3-term recurrence relation) an eigenvalue problem with a differential
operator of Dunkl-type. These polynomials can be obtained from the little
$q$-Jacobi polynomials in the limit $q=-1$. We also show that these polynomials
provide a nontrivial realization of the Askey-Wilson algebra for $q=-1$.Comment: 20 page

### Equilibria of `Discrete' Integrable Systems and Deformations of Classical Orthogonal Polynomials

The Ruijsenaars-Schneider systems are `discrete' version of the
Calogero-Moser (C-M) systems in the sense that the momentum operator p appears
in the Hamiltonians as a polynomial in e^{\pm\beta' p} (\beta' is a deformation
parameter) instead of an ordinary polynomial in p in the hierarchies of C-M
systems. We determine the polynomials describing the equilibrium positions of
the rational and trigonometric Ruijsenaars-Schneider systems based on classical
root systems. These are deformation of the classical orthogonal polynomials,
the Hermite, Laguerre and Jacobi polynomials which describe the equilibrium
positions of the corresponding Calogero and Sutherland systems. The
orthogonality of the original polynomials is inherited by the deformed ones
which satisfy three-term recurrence and certain functional equations. The
latter reduce to the celebrated second order differential equations satisfied
by the classical orthogonal polynomials.Comment: 45 pages. A few typos in section 6 are correcte

### The Bivariate Rogers-Szeg\"{o} Polynomials

We present an operator approach to deriving Mehler's formula and the Rogers
formula for the bivariate Rogers-Szeg\"{o} polynomials $h_n(x,y|q)$. The proof
of Mehler's formula can be considered as a new approach to the nonsymmetric
Poisson kernel formula for the continuous big $q$-Hermite polynomials
$H_n(x;a|q)$ due to Askey, Rahman and Suslov. Mehler's formula for $h_n(x,y|q)$
involves a ${}_3\phi_2$ sum and the Rogers formula involves a ${}_2\phi_1$ sum.
The proofs of these results are based on parameter augmentation with respect to
the $q$-exponential operator and the homogeneous $q$-shift operator in two
variables. By extending recent results on the Rogers-Szeg\"{o} polynomials
$h_n(x|q)$ due to Hou, Lascoux and Mu, we obtain another Rogers-type formula
for $h_n(x,y|q)$. Finally, we give a change of base formula for $H_n(x;a|q)$
which can be used to evaluate some integrals by using the Askey-Wilson
integral.Comment: 16 pages, revised version, to appear in J. Phys. A: Math. Theo