Some classes of generating functions for the Laguerre and 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.

We consider analytic solutions of the heat equation u xx + u yy=u t defined in a cylinder and show that any such solution can be expanded in a series of polynomial solutions to the heat equation. If we define the independent complex variables z and [zbar] by =z=x + iy, [zbar] x−y where x and y are independent complex variables, it is shown that any real-valued analytic solution of the heat equation is uniquely determined by its values on [zbar]=0 or i=0. Using this result, and expressing the above mentioned polynomial solutions to the heat equation in terms of Laguerre polynomials, we obtain some generating functions for Laguerre polynomials, as well as connection formulas between products of Hermite polynomials and Laguerre polynomials of argument r 2=x 2 + y 2. These connection formulas generalize a well known result of Feldheim.

Generalized Hermite polynomials

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.

Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big $q$-Jacobi polynomials and big $q$-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials.

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

Theory of generalized hermite polynomials

A formula expressing the Laguerre coefficients of a general-order derivative of an infinitely differentiable function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of Laguerre polynomials of any degree and for any order as a linear combination of suitable Laguerre polynomials is deduced. A formula for the Laguerre coefficients of the moments of one single Laguerre polynomial of certain degree is given. Formulae for the Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Laguerre coefficients are also obtained. A simple approach in order to build and solve recursively for the connection coefficients between Jacobi–Laguerre and Hermite–Laguerre polynomials is described. An explicit formula for these coefficients between Jacobi and Laguerre polynomials is given, of which the ultra-spherical polynomials of the first and second kinds and Legendre polynomials are important special cases. An analytical formula for the connection coefficients between Hermite and Laguerre polynomials is also obtained.

Several orthogonal polynomials have limit forms in which Hermite polynomials show up. Examples are limits with respect to certain parameters of the Jacobi and Laguerre polynomials. In this paper we are interested in more details of these limits and we give asymptotic representations of several orthogonal polynomials in terms of Hermite polynomials. In fact we give finite exact representations that have an asymptotic character. From these representations the well-known limits can be derived easily. Approximations of the zeros of the Gegenbauer polynomials $C_n^{gamma(x)$ and Laguerre polynomials $L_n^{alpha(x)$ are derived (for large values of $gamma$ and $alpha$, respectively) in terms of zeros of the Hermite polynomials and compared with numerical values. We also consider the Jacobi polynomials and the so-called Tricomi-Carlitz polynomials.

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.

Motivated by their importance and potential for applications in a variety of research fields, recently, various polynomials and their extensions have been introduced and investigated. In this sequel, we modify the known generating functions of polynomials, due to both Milne-Thomson and Dere and Simsek, to introduce a new class of generalized polynomials and present some of their involved properties. As obvious special cases of the newly introduced polynomials, we also introduce so-called power sum-Laguerre--Hermite polynomials and generalized Laguerre and Euler polynomials and we present some of their involved identities and formulas. The results presented here, being very general, are pointed out to be specialized to yield a number of known and new identities involving relatively simple and familiar polynomials.

A good portion of Gatteschi’s research publications—about 65%—is devoted to asymptotics of special functions and their zeros. Most prominently among the special functions studied figure classical orthogonal polynomials, notably Jacobi polynomials and their special cases, Laguerre polynomials, and Hermite polynomials by implication. Other important classes of special functions dealt with are Bessel functions of the first and second kind, Airy functions, and confluent hypergeometric functions, both in Tricomi’s and Whittaker’s form. This work is reviewed here, and organized along methodological lines.