346 research outputs found
A model for the continuous q-ultraspherical polynomials
We provide an algebraic interpretation for two classes of continuous
-polynomials. Rogers' continuous -Hermite polynomials and continuous
-ultraspherical polynomials are shown to realize, respectively, bases for
representation spaces of the -Heisenberg algebra and a -deformation of
the Euclidean algebra in these dimensions. A generating function for the
continuous -Hermite polynomials and a -analog of the Fourier-Gegenbauer
expansion are naturally obtained from these models
The q-harmonic oscillator and an analog of the Charlier polynomials
A model of a q-harmonic oscillator based on q-Charlier polynomials of
Al-Salam and Carlitz is discussed. Simple explicit realization of q-creation
and q-annihilation operators, q-coherent states and an analog of the Fourier
transformation are found. A connection of the kernel of this transform with
biorthogonal rational functions is observed
q-Ultraspherical polynomials for q a root of unity
Properties of the -ultraspherical polynomials for being a primitive
root of unity are derived using a formalism of the algebra. The
orthogonality condition for these polynomials provides a new class of
trigonometric identities representing discrete finite-dimensional analogs of
-beta integrals of Ramanujan.Comment: 7 pages, LATE
Completeness of Coherent States Associated with Self-Similar Potentials and Ramanujan's Integral Extension of the Beta Function
A decomposition of identity is given as a complex integral over the coherent
states associated with a class of shape-invariant self-similar potentials.
There is a remarkable connection between these coherent states and Ramanujan's
integral extension of the beta function.Comment: 9 pages of Late
Discrete Darboux transformation for discrete polynomials of hypergeometric type
Darboux Transformation, well known in second order differential operator
theory, is applied here to the difference equation satisfied by the discrete
hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn)
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
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
New connection formulae for the q-orthogonal polynomials via a series expansion of the q-exponential
Using a realization of the q-exponential function as an infinite
multiplicative sereis of the ordinary exponential functions we obtain new
nonlinear connection formulae of the q-orthogonal polynomials such as
q-Hermite, q-Laguerre and q-Gegenbauer polynomials in terms of their respective
classical analogs.Comment: 14 page
Asymptotics and Dimensional Dependence of the Number of Critical Points of Random Holomorphic Sections
We prove two conjectures from [M. R. Douglas, B. Shiffman and S. Zelditch,
Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics.
J. Differential Geom. 72 (2006), no. 3, 381-427] concerning the expected number
of critical points of random holomorphic sections of a positive line bundle. We
show that, on average, the critical points of minimal Morse index are the most
plentiful for holomorphic sections of {\mathcal O}(N) \to \CP^m and, in an
asymptotic sense, for those of line bundles over general K\"ahler manifolds. We
calculate the expected number of these critical points for the respective cases
and use these to obtain growth rates and asymptotic bounds for the total
expected number of critical points in these cases. This line of research was
motivated by landscape problems in string theory and spin glasses.Comment: 14 pages, corrected typo
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