Multivariate Hermite Polynomials Research Articles

Tomography of multimode quantum systems with quadratic Hamiltonians and multivariable Hermite polynomials

The systems with multimode nonstationary Hamiltonians quadratic in position and momentum operators are reviewed. The tomographic probability distributions (tomograms) for the Fock states and Gaussian states of the quadratic systems are discussed. The tomograms for the Fock states are expressed in terms of multivariable Hermite polynomials. Using the obvious physical relations some new formulas for multivariable Hermite polynomials are found. Examples of oscillator and charge moving in electromagnetic field are presented.

Generalized polynomials, operational identities and their applications

It is shown that an appropriate combination of methods, relevant to generalized operational calculus and to special functions, can be a very useful tool to treat a large body of problems both in physics and mathematics. We discuss operational methods associated with multivariable Hermite, Laguerre, Legendre, and other polynomials to derive a wealth of identities useful in quantum mechanics, electromagnetism, optics, etc., or to derive new identities between special functions as, e.g., Mehler- or mixed-type generating functions.

A simple expression for the multivariate Hermite polynomials

We give a simple method for obtaining the multivariate Hermite polynomials. Explicitly we give all of them up to order three: these are needed for the second-order Edgeworth expansions for the distribution and density of most standardised vector estimates.

Intertwining operators for a degenerate double affine Hecke algebra and multivariable orthogonal polynomials

Several properties of the multivariable Hermite and Laguerre polynomials associated with the quantum Calogero models are investigated by using the operators that intertwine representations of a degenerate version of the double affine Hecke algebra. As applications, raising operators and shift operators for the polynomials are constructed in a unified manner.

Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators

Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on $\b R^N$. The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In case of the symmetric group $S_N$, our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments.

A Simple Derivation of Multivariable Hermite and Legendre Polynomials

Multivariable Hermite and Legendre polynomials are derived by a simple perturbative procedure. Both polynomials solve eigenvalue equations for quantum many-body problems with long-range interaction.

Optical transformer of photon statistics

The possibility of a linear optical transformer of photon statistics for multimode light is suggested. The transformation coefficients are obtained explicitly in terms of multivariable Hermite polynomials. As physical examples of a linear transformer a resonator with moving walls and a lossless beam splitter are discussed as optical elements that change the photon statistics of light beams. The beam splitter is shown to produce redistribution of quadrature squeezing and correlation between two modes of light.

The d-variate vector hermite polynomial of order k

The multivariate Hermite polynomials occur in Edgeworth and saddle-point approximations to the distributions of random vector variables. Explicit expressions in vector notation for the multivariate Hermite polynomials of arbitrary order are given and shown to be a special case of a general addition property of Hermite polynomials. General properties of these polynomials are given in vector notation, including an integral representation, the orthogonality property, differential properties, and recurrence relations as well as inversion formulas. Various expansions of a multivariate normal density are given in terms of a multivariate normal density with different mean, different covariance, or both. In special cases these results give Mehler's formula.

The d-variate vector Hermite polynomial of order k

Abstract The multivariate Hermite polynomials occur in Edgeworth and saddle-point approximations to the distributions of random vector variables. Explicit expressions in vector notation for the multivariate Hermite polynomials of arbitrary order are given and shown to be a special case of a general addition property of Hermite polynomials. General properties of these polynomials are given in vector notation, including an integral representation, the orthogonality property, differential properties, and recurrence relations as well as inversion formulas. Various expansions of a multivariate normal density are given in terms of a multivariate normal density with different mean, different covariance, or both. In special cases these results give Mehler's formula.

Fourier expansions and multivariable bessel functions concerning radiation problems

The link between generalized Bessel functions and other special functions is investigated using the Fourier series and the generalized Jacobi-Anger expansion. A new class of multivariable Hermite polynomials is then introduced and their relevance to physical problems discussed. As an example of the power of the method, applied to radiation physics, we analyse the role played by multi-variable Bessel functions in the description of radiation emitted by a charge constrained to a nonlinear oscillation.

Even and odd coherent states for multimode parametric systems.

The multimode even and odd coherent states (multimode Schr\"odinger cat states) are constructed for polymode parametric oscillators of the electromagnetic field. The evolution of the photon distribution function is evaluated explicitly. The distribution function is expressed in terms of multivariable Hermite polynomials; the means and dispersions of the function are calculated. The conditions for the existence of squeezing are formulated. The correlations between the different modes of the Schr\"odinger cat states are studied. The transformation of the initial Schr\"odinger cat states under the action of a resonant external force is investigated.

Numerical Solution of Continuous-State Dynamic Programs Using Linear and Spline Interpolation

This paper demonstrates that the computational effort required to develop numerical solutions to continuous-state dynamic programs can be reduced significantly when cubic piecewise polynomial functions, rather than tensor product linear interpolants, are used to approximate the value function. Tensor product cubic splines, represented in either piecewise polynomial or B-spline form, and multivariate Hermite polynomials are considered. Computational savings are possible because of the improved accuracy of higher-order functions and because the smoothness of higher-order functions allows efficient quasi-Newton methods to be used to compute optimal decisions. The use of the more efficient piecewise polynomial form of the spline was slightly superior to the use of Hermite polynomials for the test problem and easier to program. In comparison to linear interpolation, use of splines in piecewise polynomial form reduced the CPU time to obtain results of equivalent accuracy by a factor of 250–330 for a stochastic 4-dimensional water supply reservoir problem with a smooth objective function, and factors ranging from 25–400 for a sequence of 2-, 3-, 4-, and 5-dimensional problems. As a result, a problem that required two hours to solve with linear interpolation was solved in a less than a minute with spline interpolation with no loss of accuracy.

Multivariate Hermite Expansion of Hydrodynamic Drag Loads on Tension Leg Platforms

In this paper, a new method is proposed for the expansion of nonlinear drag forces in terms of multivariate Hermite polynomials correct up to the second order. The drag‐force formulation includes the effect of instantaneous wave surface profile and it caters for the waves and currents approaching from any arbitrary direction with respect to the platform orientation. These attributes are critical for a reliable treatment of the wave‐induced viscous effects on tension leg platforms (TLP). The viscous nonlinear drag force expressed in terms of Hermite polynomials is decomposed into the mean (zeroth‐order), viscous exciting and viscous damping terms (first‐order) and the slowly varying drift‐force term (second‐order). This decomposition permits spectral representation of the first‐order viscous forces in terms of the spectral density function of the water particle velocities. Accordingly, the second‐order viscous force can be described within the spectral framework by the spectral convolution or other related...

A chain rule for differentiation with applications to multivariate hermite polynomials

A chain rule is given for differentiating a multivariate function of a multivariate function. In the univariate case this chain rule reduces to Faa de Bruno's formula.Using this, a simple procedure is given to obtain the rth order multivariate Hermite polynomial from the rth order univariate Hermite polynomial.