13 research outputs found
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 . The proof
of Mehler's formula can be considered as a new approach to the nonsymmetric
Poisson kernel formula for the continuous big -Hermite polynomials
due to Askey, Rahman and Suslov. Mehler's formula for
involves a sum and the Rogers formula involves a sum.
The proofs of these results are based on parameter augmentation with respect to
the -exponential operator and the homogeneous -shift operator in two
variables. By extending recent results on the Rogers-Szeg\"{o} polynomials
due to Hou, Lascoux and Mu, we obtain another Rogers-type formula
for . Finally, we give a change of base formula for
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
On a q-extension of Mehta's eigenvectors of the finite Fourier transform for q a root of unity
It is shown that the continuous q-Hermite polynomials for q a root of unity
have simple transformation properties with respect to the classical Fourier
transform. This result is then used to construct q-extended eigenvectors of the
finite Fourier transform in terms of these polynomials.Comment: 12 pages, thoroughly rewritten, the q-extended eigenvectors now
N-periodic with q an M-th root of
Coherent states for the hydrogen atom: discrete and continuous spectra
We construct the systems of generalised coherent states for the discrete and
continuous spectra of the hydrogen atom. These systems are expressed in
elementary functions and are invariant under the (discrete spectrum)
and (continuous spectrum) subgroups of the dynamical symmetry group
of the hydrogen atom. Both systems of coherent states are particular
cases of the kernel of integral operator which interwines irreducible
representations of the group.Comment: 15 pages, LATEX, minor sign corrections, to appear in J.Phys.
A Non-Standard Generating Function for Continuous Dual -Hahn polynomials
We study a non-standard form of generating function for the three-parameter continuous dual q-Hahn polynomials , which has surfaced in a recent work of the present authors on the construction of lifting -difference operators in the Askey scheme of basic hypergeometric polynomials. We show that the resulting generating function identity for the continuous dual q-Hahn polynomials can be explicitly stated in terms of Jackson’s -exponential functions .Estudiamos una forma no estándar de la función generatriz para una familia de polinomios duales continuos -Hahn de tres parámetros , que han surgido en un trabajo reciente de los autores en la construcción de operadores elevadores en -diferencias del esquema de Askey de polinomios básicos hipergeométricos. Demostramos que la función generatriz identidad resultante para los polinomios q-Hahn duales continuos puede ser expresada explÃcitamente en términos de las funciones -exponenciales de Jackson
More on algebraic properties of the discrete Fourier transform raising and lowering operators
In the present work, we discuss some additional findings concerning algebraic properties of the N-dimensional discrete Fourier transform (DFT) raising and lowering difference operators, recently introduced in [Atakishiyeva MK, Atakishiyev NM (2015), J Phys: Conf Ser 597, 012012; Atakishiyeva MK, Atakishiyev NM (2016), Adv Dyn Syst Appl 11, 81–92]. In particular, we argue that the most authentic symmetrical form of discretization of the integral Fourier transform may be constructed as the discrete Fourier transforms based on the odd points N only, while in the discrete Fourier transforms on the even points N this symmetry is spontaneously broken. This heretofore undetected distinction between odd and even dimensions is shown to be intimately related with the newly revealed algebraic properties of the above-mentioned DFT raising and lowering difference operators and, of course, is very consistent with the well-known formula for the multiplicities of the eigenvalues, associated with the N-dimensional DFT. In addition, we propose a general approach to deriving the eigenvectors of the discrete number operators
N(N)
N(N)
, that avoids the above-mentioned pitfalls in the structure of each even-dimensional case N = 2L