Generalized Hermite Polynomials Research Articles

Generalized Bessel Functions and Generalized Hermite Polynomials

In this note we show that generalized Hermite polynomials of the Gould-Hopper type are linked to the multivariables generalized Bessel functions. We show that such a formalism encompasses both first and second kind Gould-Hopper polynomials and that their recurrence properties are straightforwardly derived from those of generalized Bessel functions

Wavelet array decomposition of images using a Hermite sieve

Generalized Hermite polynomials are used in a novel way to arrive at a multi-layered representation of images. This representation, which is centred on the creation of a new class ofwavelet arrays, is (i) distinct from what we find in the current literature, (ii) stable, and (iii) in the manner of standard transforms, transforms the image, explicitly, into matrices of coefficients, reminiscent of Fourier series,but at various scales, controlled by ascale parameter. Among the other properties of the wavelet arrays, (a) the shape of the resolution cell in the ‘phase-space’ is variable even at a specified scale, depending on the nature of the signal under consideration; and (b) a systematic procedure is given for extracting the zero-crossings from the coefficients at various scales. This representation has been successfully applied to both synthetic and natural images, including textures.

The Product Formula and Convolution Structure Associated with the Generalized Hermite Polynomials

Here the product formula for the generalized and suitably normalized Hermite polynomials with parameter μ ≥ 0 will be explicitly established. Its measure turns out to be absolutely continuous and supported on two disjoint intervals lying symmetrically on the real line, provided that μ > 0. In the limit case μ = 0, which is associated with the classical Hermite polynomials, four additional point masses occur at the endpoints of the two intervals. As an application, the product formula is used to introduce a generalized translation operator and a corresponding convolution product on appropriately weighted Lebesgue spaces. To this end, norm estimates of the translation operator from above and below are presented. For any μ ≥ 1 2 , this gives rise to a quasi-positive convolution algebra.

Squeezed states, generalized Hermite polynomials and pseudo-diffusion equation

We show that the wavefunctions 〈 pq; λ| n〈, of the harmonic oscillator in the squeezed state representation, have the generalized Hermite polynomials as their natural orthogonal polynomials. These wavefunctions lead to generalized Poisson Distribution P n ( pq;λ), which satisfy an interesting pseudo-diffusion equation: ∂P np,q;λ) ∂λ = 1 4 [ ∂ 2 ∂p 2 −( 1 λ 2 ) ∂ 2 ∂q 2 ]P 2(p,q;λ) , in which the squeeze parameter λ plays the role of time. Th entropies S n (λ) have minima at the unsqueezed states (λ=1), which means that squeezing or stretching decreases the correlation between momentum p and position q.

Hermite polynomials for signal reconstruction from zero-crossings. Part 1: one-dimensional signals

Generalised Hermite polynomials are employed for the reconstruction of an unknown signal from a knowledge of its zero-crossings, under certain conditions on its spatial/spectral width, but dispensing with the assumption of bandlimitedness. A computational implementation of the proposed method is given for one-variable (or one-dimensional) signals, featuring an application of simulated annealing for optimal reconstruction.

The quantum relativistic harmonic oscillator: generalized Hermite polynomials

A relativistic generalisation of the algebra of quantum operators for the harmonic oscillator is proposed. The wave functions are worked out explicitly in configuration space. Both the operator algebra and the wave functions have the appropriate c→∞ limit. This quantum dynamics involves an extra quantization condition mc 2/ωℏ = 1, 3 2 , 2 , … of a topological character.

Sur la suite de polynômes orthogonaux associée à la formeu=δ c +λ(x−c) −1 L

We show that, ifL is regular, semi-classical functional, thenu is also regular and semi-classical for every complex λ, except for a discrete set of numbers depending onL andc. We give the second order linear differential equation satisfied by each polynomial of the orthogonal sequence associated withu. The cases whereL is either a classical functional (Hermite, Laguerre, Bessel, Jacobi) or a functional associated with generalized Hermite polynomials are treated in detail.

Nonlinear diffusion and density functional theory

A phenomenological dynamic extension of the classical density functional theory is proposed. Our evolution equation for the density has the form of a generalized Smoluchowski equation which involves a correlation potential to be derived from the microscopic theory. By means of an expansion in terms of generalized Hermite polynomials we also present numerical results for simple cases of nonlinear diffusion.

Special functions in Clifford analysis and axial symmetry

In our previous paper ( Rend. Circ. Mat. Palermo 6 (1984), 259–269, we proved a general Laurent expansion for monogenic functions in symmetric domains of R m + 1 , depending on the kind of symmetry involved. In this paper we consider axial symmetric domains and we apply our results in order to introduce new special monogenic functions. We investigated axial exponential functions, generalized powerfunctions and generalized Hermite polynomials.

Exact solutions of unitarily invariant matrix models in zero dimensions

Explicit constructions of Green functions for U(N) invariant-matrix Φ4 theories are given in zero space-time dimensions. The results are expressed in terms of a new class of orthogonal polynomials, which are orthogonal on the interval [−∞,∞] with respect to the weight function exp{−m2φ2−λφ4}. An explicit construction of the coefficients of these generalized Hermite polynomials is presented. We briefly discuss the ‘‘thermodynamic’’ limit N→∞ of the corresponding one-dimensional statistical system of N classical particles with logarithmic pair interactions and subjected to an external anharmonic potential.

A kinetic theory of dense fluids. Test-particle kinetic equation

Mori’s form of the generalized Langevin equation is applied to a system of particles whose pair interactions are sums of short-ranged, rigid-core repulsions and longer-ranged continuous interactions. It is assumed that the memory function decays exponentially from its initial value at a rate characterized by a single, wave number dependent relaxation time. The test–particle kinetic equation resulting from this one key assumption is strongly reminiscent of that associated with the well-known Rice–Allnatt theory. However, in addition to Fokker–Planck and Enskig operators similar to those of the Rice–Allnatt theory, it also includes coupling between rigid-core scattering events and the diffusive, Brownian motions of the particles as well as contributions associated with three-particle rigid-core collisions. A systematic method of solution is developed and applied to a simplified form of this kinetic equation. Solutions generated by this procedure involve matrix elements of the linearized Enskog operator computed in the basis of generalized Hermite polynomials which are eigenfunctions of the Fokker–Planck operator associated with the long-ranged, continuous portion of the pair potential. The approximate solutions obtained are used to compute the density and velocity autocorrelation functions and the coefficient of self-diffusion as well.

Some Properties and Applications of the Differences of the Generalized Factorials

The higher order difference $[ {\mathop \Delta \limits_a ^n ( x )_{m,b} } ]_{x = 0} $ of the generalized factorial $( x )_{m,b} = x( {x - b} )( {x - 2b} ) \cdots ( {x - mb + b} )$, where the difference operator $\mathop \Delta \limits_a $ is defined by $\mathop \Delta \limits_a f( x ) = f( {x + a} ) - f( x )$, is the subject of this paper. The numbers $p( {m,n;a,b} ) = ( {b^{ - m} / n!} )[ {\mathop \Delta \limits_a ^n ( x )_{m,b} } ]_{x = 0} $ used by Jordan [12, p. 221;1960] appeared to be the same as the numbers $C( {m,n,s} ) = ( {1 / n!} )[ {\Delta ^n ( {sx} )_m } ]_{x = 0} ,s = {a / b}$, studied by the author [5; 1975], [6;1976], [7; 1977]. Some additional properties and applications of these numbers are given here.More specifically, some expansions by aid of the numbers $C( {m,n,s} )$ useful in occupancy theory are developed and a representation of $C( {m,m - n,s} )$ as sum of factorials is obtained. The coefficients of this representation have an interesting combinatorial interpretation. Some applications in occupancy theory are briefly indicated. Finally $C( {m,n,s} )$ are shown to be the coefficients of the generalized Hermite polynomials.

Some Classes of Generating Functions for the Laguerre and 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.

Some classes of generating functions for the Laguerre and 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.

A Note on a Generating Function for the Generalized Hermite Polynomials

Several generating-function relations involving the polynomials {ie1}, and their natural generalization {ie2}, are discussed. A hitherto seemingly unnoticed fact on the equivalence of certain known generating functions is also pointed out.

Generalized Hermite polynomials

Hermite polynomials of several variables are defined by a generalization of the Rodrigues formula for ordinary Hermite polynomials. Several properties are derived, including the differential equation satisfied by the polynomials and their explicit expression. An application is given.

Generalized Hermite polynomials

Hermite polynomials of several variables are defined by a generalization of the Rodrigues formula for ordinary Hermite polynomials. Several properties are derived, including the differential equation satisfied by the polynomials and their explicit expression. An application is given.

A Generalized Hermite Distribution and Its Properties

A generalized Hermite distribution with an additional parameter is defined in terms of generalized Hermite polynomials. Various relations among these polynomials are given. The parameters in the mean and the variance of the distribution are related in such a way that the variance increases or decreases faster than the mean.

Some polynomials associated with the generalized Hermite polynomials

Some polynomials associated with the generalized Hermite polynomials

Erratum: “Tables of zeros and Gaussian weights of certain associated Laguerre polynomials and the related generalized Hermite polynomials”

Erratum: “Tables of zeros and Gaussian weights of certain associated Laguerre polynomials and the related generalized Hermite polynomials”