A Hahn-Type Characterization of Generalized Hermite Polynomials Through a Dunkl-Based Raising Operator

We investigate the symmetric Dunkl-classical orthogonal polynomials by using a new approach in connection with the Dunkl operator. The main aim of this technique is first and foremost to determine the recurrence coefficients. We establish the existence and uniqueness of symmetric Dunkl-classical orthogonal polynomials, and confirm that the two families of generalized Hermite orthogonal polynomials and Gegenbauer orthogonal polynomials are the only ones belonging to this class. Two apparently new characterizations in a more general setting are given. This paper complements earlier work of Ben Cheikh and Gaied [Characterization of the Dunkl-classical symmetric orthogonal polynomials. Appl Math Comput. 2007;187:105–114. doi: 10.1016/j.amc.2006.08.108].

In this paper, we consider the following \(\mathcal{L}\)-difference equation$$\Phi(x) \mathcal{L}P_{n+1}(x)=(\xi_nx+\vartheta_n)P_{n+1}(x)+\lambda_nP_{n}(x),\quad n\geq0,$$where \(\Phi\) is a monic polynomial (even), \(\deg\Phi\leq2\), \(\xi_n,\,\vartheta_n,\,\lambda_n,\,n\geq0\), are complex numbers and \(\mathcal{L}\) is either the Dunkl operator \(T_\mu\) or the the \(q\)-Dunkl operator \(T_{(\theta,q)}\). We show that if \(\mathcal{L}=T_\mu\), then the only symmetric orthogonal polynomials satisfying the previous equation are, up a dilation, the generalized Hermite polynomials and the generalized Gegenbauer polynomials and if \(\mathcal{L}=T_{(\theta,q)}\), then the \(q^2\)-analogue of generalized Hermite and the \(q^2\)-analogue of generalized Gegenbauer polynomials are, up a dilation, the only orthogonal polynomials sequences satisfying the \(\mathcal{L}\)-difference equation.

The paper gives a review of recent progress in the classification of monodromy-free Schr\"odinger operators with rational potentials. We concentrate on a class of potentials constituted by generalized Hermite polynomials. These polynomials defined as Wronskians of classic Hermite polynomials appear in a number of mathematical physics problems as well as in the theory of random matrices and 1D SUSY quantum mechanics. Being quadratic at infinity, those potentials demonstrate localized oscillatory behavior near the origin. We derive an explicit condition of non-singularity of the corresponding potentials and estimate a localization range with respect to indices of polynomials and distribution of their zeros in the complex plane. It turns out that 1D SUSY quantum non-singular potentials come as a dressing of the harmonic oscillator by polynomial Heisenberg algebra ladder operators. To this end, all generalized Hermite polynomials are produced by appropriate periodic closure of this algebra which leads to rational solutions of the Painlev\'e IV equation. We discuss the structure of the discrete spectrum of Schr\"odinger operators and its link to the monodromy-free condition.

Special polynomials (e.g. Hermite polynomials) are very important for the development of physics and mathematics. As a further extension of ordinary Hermite polynomials, we introduce new generalized Hermite polynomials with three variables and find their generating functions using the operator ordering method in quantum optics. Also, some new operator identities and integral formulas are obtained. As applications, the normalization, Wigner functions and evolutions for certain quantum states are analytically presented. These analytical results can provide conveniences for numerically studying the properties and applications of such quantum states.

We study the Hermite interpolation problem on the spaces of symmetric bivariate polynomials. We show that the multipoint Berzolari-Radon sets solve the problem. We also give a Newton formula for the interpolation polynomial and use it to prove a continuity property of the interpolation polynomial with respect to the interpolation points.

This paper studies a suitably normalized set of generalized Hermite polynomials and sets down a relevant Mehler formula, Rodrigues formula, and generalized translation operator. Weighted generalized Hermite polynomials are the eigenfunctions of a generalized Fourier transform which satisfies an F. and M. Riesz theorem on the absolute continuity of analytic measures. The Bose-like oscillator calculus, which generalizes the calculus associated with the quantum mechanical simple harmonic oscillator, is studied in terms of these polynomials.

Model of the extended Dunkl oscillator based on Milovanović generalized Hermite polynomials on radial rays is discussed. Simple explicit realization of creation and annihilation operators in terms of difference-differential operators, coherent states are investigated.

Generalized Hermite polynomials

Theory of generalized hermite polynomials

In the first half of the article, we present two theorems which give, as special cases, a number of new classes of generating functions for the Laguerre polynomial. These formulae extend the recent results of Carlitz [2] and others. The latter part of our work deals with two theorems involving new generating functions for the Hermite and generalized Hermite polynomials, thus generalizing some well-known expansions. The method of proof adopted in this paper differs from that of previous workers.

The associated Laguerre polynomials LH"(x) and the Hermite polynomials #*(£) occupy a distinctive place in quantum mechanics.A wellestablished relationship exists between Hk(£) and Ln±'b(x); this can be further exploited to obtain generalized Hermite polynomials Hk*(%).The latter, relevant to spatial harmonic oscillators in curvilinear coordinates, are essentially the polynomials obtained by Stroud and Secrest [11] based on transformations of numerical quadratures.The tables presented here are zeros and Gaussian weights of {Ln"(x)\ with a = -0.5(1.0)3.5, re = 4, 8, 16, 32; and those of {#/(!)! with A = 0(1.0)4.0,k = 8, 16, 32, 64.In both cases the results are believed to be correct to one unit in the 25th decimal place.The notations for Lna(x) and Hk(%) employed here are those in Szegö [12] and adopted in Erdélyi et al. [5]; namely a» ^w-("íj.«-»;«+^)-¿(:í:)(x)m/ml, lk/2] (1.2) #*({) = fc!E (-l)m(2£)*-27m!(fc -2m)! \Lna{x)} thus defined forms an orthogonal set in [0, » ) with respect to the weight function e~xx"; Hk(£) are similarly orthogonal in ( -», + oo ) with respect to e~£ .Many physicists, notably Condon and Shortley [3], employ (1.3) Lna(x) = re! ( j iPj( -re + a; a + 1; x) = a polynomial of degree n -a, while many others, e.g., Morse and Feshbach [8] have (1.4)Ln"(x) = (" + «)! (n + a) iFi{-n; a + 1; x).The mathematical statistics literature often employs the notation [km (1.5) Hk(S) = 2~*/2fc!£ (-l)m(V2H)k-2m/ml (fc -2m)! which would correspond to Hek(i) in Erdélyi et al., and /&*(£) in Davis [4], and the corresponding weighting function is e_i /2.

Hermite polynomials with many variables and many indices play a crucial role within the framework of phase-space formulation of classical or quantum mechanics. A generalized operational formalism useful to handle these polynomials is discussed and also a new set of creation–annihilation operators associated to the phase-space harmonic oscillator orthogonal functions is introduced. The Fourier transform of these orthogonal functions is studied by discussing the analogies with the ordinary case. Finally, the integral representation of the generalized Hermite polynomials and a simple technique to deal with the generalized heat equation are discussed.

Generalized Hermite polynomials in superspace as eigenfunctions of the supersymmetric rational CMS model

The Painlevé-IV equation has three families of rational solutions generated by the generalized Hermite polynomials. Each family is indexed by two positive integers $m$ and $n$. These functions have applications to nonlinear wave equations, random matrices, fluid dynamics, and quantum mechanics. Numerical studies suggest the zeros and poles form a deformed $n\times m$ rectangular grid. Properly scaled, the zeros and poles appear to densely fill certain curvilinear rectangles as $m,n\to \infty $ with $r:=m/n$ a fixed positive real number. Generalizing a method of Bertola and Bothner [2] used to study rational Painlevé-II functions, we express the generalized Hermite rational Painlevé-IV functions in terms of certain non-Hermitian orthogonal polynomials. Using the Deift–Zhou nonlinear steepest-descent method, we asymptotically analyze the associated Riemann–Hilbert problem in the limit $n\to \infty $ with $m=r\cdot n$ for $r$ fixed. We obtain an explicit characterization of the boundary curve and determine the leading-order asymptotic expansion of the rational Painlevé-IV functions associated to generalized Hermite polynomials in the pole-free region.

Summation formulae for Gould–Hopper generalized Hermite polynomials