In this paper, we introduce the notion of Dunkl-classical orthogonal polynomials. Then, we show that generalized Hermite and generalized Gegenbauer polynomials are the only Dunkl-classical symmetric orthogonal polynomials by solving a suitable differential-difference equation. 2006 Elsevier Inc. All rights reserved.

This paper concerns the Schur stability of polynomials. Firstly necessary conditions are presented for the Schur stability of complex polynomials with fixed coefficients based on the discrete Hermite-Biehler theorem. Then the result is applied to obtain necessary conditions for the robust Schur stability of real interval polynomials.

The asymptotic behavior of quadratic Hermite-Padé polynomials pn, qn, rn ∈ Pn of type I and pn, qn, rn ∈ P2n of type II associated with the exponential function are studied. In the introduction the background of the definition of Hermite-Padé polynomials is reviewed. The quadratic Hermite-Padé polynomials pn, qn, rn ∈ Pn of type I are defined by the relation pn(z) + qn(z)e z + rn(z)e 2z = O(z) ...

We show that the equilibrium positions of the Ruijsenaars-Schneider-van Diejen systems with the trigonometric potential are given by the zeros of the Askey-Wilson polynomials with five parameters. The corresponding single particle quantum version, which is a typical example of “discrete” quantum mechanical systems with a q-shift type kinetic term, is shape invariant and the eigenfunctions are t...

A series expansion with remainder for functions in a Sobolev space is derived in terms of the classical Bernoulli polynomials, the B-spline scale-space and the continuous wavelet transforms with the derivatives of the standardized B-splines as mother wavelets. In the limit as their orders tend to infinity, the B-splines and their derivatives converge to the Gaussian function and its derivatives...

We show that the formalism of hybrid polynomials, interpolating between Hermite and Laguerre polynomials, is very useful in the study of Motzkin numbers and central trinomial coefficients. These sequences are identified as special values of hybrid polynomials, a fact which we use to derive their generalized forms and new identities satisfied by them.

We show that the Meixner, Pollaczek, Meixner-Pollaczek and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures. We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials. Running Title: Generating Functions

A q-version of the Fourier transformation and some of its properties are discussed. INTRODUCTION Models of q-harmonic oscillators are being developed in connection with quantum groups and their various applications (see, for example, Refs. [M], [Bi], [AS1], and [AS2]). For a complete correspondence with the quantum-mechanical oscillator problem, these models need an analog of the Fourier transf...

The expansion of quantum states and operators in terms Fock plays a fundamental role the field continuous-variable mechanics. In particular, for general single-mode Gaussian noisy states, many different approaches have been used evaluation their representation. this paper, natural approach has applied using exclusively operational properties Hermite Hermite-like polynomials showing field. Close...

— We extend a randomisation method, introduced by Shiffman-Zelditch and developed by Burq-Lebeau on compact manifolds for the Laplace operator, to the case of R with the harmonic oscillator. We construct measures, thanks to probability laws which satisfy the concentration of measure property, on the support of which we prove optimal weighted Sobolev estimates on R. This construction relies on a...