Theory of generalized hermite polynomials

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.

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

We use the Gould-Hopper (GH) polynomials to investigate the Novikov-Veselov (NV) equation. The root dynamics of the σ-flow in the NV equation is studied using the GH polynomials and then the Lax pair is found. In particular, when N = 3, 4, 5, one can get the Gold-fish model. The smooth rational solutions of the NV equation are also constructed via the extended Moutard transformation and the GH polynomials. The asymptotic behavior is discussed and then the smooth rational solution of the Liouville equation is obtained.

Expansions in series of functions are one of the most important tools of the applied mathematician, particularly expansions in series of the classical orthogonal polynomials, e.g. Laguerre, Jacobi and Hermite polynomials. In applied problems, the uniqueness of the particular expansion is usually intrinsic to the analysis, and often implicitly assumed. Indeed, in those cases where the functions in the series are orthogonal, uniqueness can often be proved by an argument that runs as follows. Let {φn(x)} (n = 0, 1, 2, …) be a sequence of functions orthogonal with respect to the weight function ρ(x) over the interval [0, 1], and suppose thatthe series being boundedly convergent for 0 ≤ x ≤ 1.

The Weber-Hermite differential equation, obtained as the dimensionless form of the stationary Schroedinger equation for a linear harmonic oscillator in quantum mechanics, has been expressed in a generalized form through introduction of a constant conjugation parameter according to the transformation , where the conjugation parameter is set to unity () at the end of the evaluations. Factorization in normal order form yields composite eigenfunctions, Hermite polynomials and corresponding positive eigenvalues, while factorization in the anti-normal order form yields the partner composite anti-eigenfunctions, anti-Hermite polynomials and negative eigenvalues. The two sets of solutions are related by an reversal conjugation rule . Setting provides the standard Hermite polynomials and their partner anti-Hermite polynomials. The anti-Hermite polynomials satisfy a new differential equation, which is interpreted as the conjugate of the standard Hermite differential equation.

Special functions in Clifford analysis and axial symmetry

The dynamic response of lattice-type structures, assuming small displacements, may be described by the following set of four matrix equations: (1) Equation of consistent displacements; (2) joint equation of equilibrium; (3) force displacement equation; and (4) equation of support. The response function or indicial flexibility matrix is formed of terms that are solutions of the partial differential equations describing the transverse, torsional, and longitudinal vibrations of uniform, slender beams. The set of matrix equations becomes a set of linear, algebraic equations when transformed to the Laplace domain. Solutions that are carried out in the Laplace domain yield expressions for displacements, rotations, end moments, and forces in terms of the transform variable. Inversions are carried out in terms of series expansions of orthogonal functions. External damping, together with forcing function cut-off, is introduced to assure the uniform convergence of the series for all values of time. The application of the equations to the externally damped cantilever beam and to the unsymmetrical portal verifies the validity of the method.

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.

Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice