Hilbert transforms of resonance functions

Expressions have been derived for the Hilbert transforms of resonance functions for induction and dispersion forces in spectral line-shape theories. These functions are evaluated numerically and are also represented by power series.     

Inverse Laplace transforms of osculatory and hyperosculatory interpolation polynomials

In the numerical calculation of f(t), the inverse Laplace transform of F(p), where f(′) = (1/2πi) °_c−i∞^c+i∞ e^pt F(p)dp, sufficient accuracy is usually obtainable when p³F(p), s > 0, is replaced by an interpolating polynomial in 1/p. From the values of F(p) with F′(p), or with F′(p) and F″(p), for p at points equally spaced on the real axis, an osculatory or hyperosculatory interpolation polynomial for p⁸F(p), namely L_2n−1(x) or L_3n−1(x), where x = 1/p, is obtained in barycentric form. Then f(t) is calculated by a Gaussian-type quadrature formula employing complex values of L_2n−1 or L_3n−1 and instead of p^sF(p) which may be unknown or more difficult to compute. For calculating L_2n−1 and L_3n−1, auxiliary coefficients, suitable for economical storage in the program, are given exactly for n = 2(1)11 and n = 2(1)7, furnishing up to 21st and 20th degree accuracy, respectively.     

Lie-theoretic generating relations of two variable Laguerre polynomials

Generating relations involving two variable laguerre polynomials L_n(x, y) are derived. The process involves the construction of a three-dimensional Lie algebra isomorphic to special linear algebra sl(2) with the help of Weisner's method by giving suitable interpretations to the index n of the polynomials L_n(x, y).     

Generating functions for Hermite polynomials of arbitrary order

We propose a systematic method for the construction of generating functions for Hermite polynomials of arbitrary order. The procedure is based on a suitable formula for the Hermite polynomials and our results contain ones obtained earlier by Nieto and Truax [Phys. Lett. A 208 (1995) 8] as particular cases.     

The generating functional for the probability density functions of Navier-Stokes turbulence

A generating functional for the equal-time spatial probability density functions which represent the ensemble of turbulent incompressible Navier-Stokes fluids is introduced. By formally solving the linear evolution equation satisfied by this functional, the probability densities are represented as functional integrals. It is shown that the generating functional can be regarded as the space characteristic functional of a generalized random field defined on the phase space spanned by the material position and velocity fields of a fluid particle. The interpretation of this random field, which satisfies a dynamical equation similar to Vlasov's, is clarified through the formal analogies between the statistics of molecules and fluid particles at the functional level. A class of statistically realizable and solvable models is also considered within the context of the present formalism.     

On integral and finite Fourier transforms of continuous <Emphasis Type="Italic">q</Emphasis>-Hermite polynomials

We give an overview of the remarkably simple transformation properties of the continuous q-Hermite polynomials H _n (x|q) of Rogers with respect to the classical Fourier integral transform. The behavior of the q-Hermite polynomials under the finite Fourier transform and an explicit form of the q-extended eigenfunctions of the finite Fourier transform, defined in terms of these polynomials, are also discussed. The text was submitted by the authors in English.     

On generating functions of higher-spin cubic interactions

We present off-shell generating functions for all cubic interactions of totally symmetric massless Higher-Spin gauge fields and discuss their properties.     

On Laguerre polynomials and the convolution product of parabolic cylinder functions

We compute the convolution product of a pair of parabolic cylinder functions of integer non-negative indices and express the result in terms of Laguerre polynomials. Symmetry properties, recurrence formulas and physical applications of such products as well as some of their generalizations are also investigated.     

The method of generating functions for the calculation of matrix elements in the deformed-harmonic oscillator basis

The method of generating functions which was previously only employed for the spherical basis of harmonic-oscillator single-particle wave functions is here generalized to the deformed (=cylindrical = asymptotic) basis. One-center and two-center matrix elements which are important in fission or heavy-ion scattering theories are obtained for various operators and potentials, i.e., spin-orbit, l-squared, Gaussian, or Gaussian multiplied by a polynomial, etc. If they cannot be calculated explicitely, recurrence formulae are evaluated. The method circumvents integrating the matrix elements by the introduction of generating parameters and looking for the coefficient of some power of the generating parameter in the expansion of the generating function. The method is further simplified by transforming the operators acting on the wave functions into operators which act onto the generating function or into functions to be multiplied by the generating integral.     

Next-to-leading-order QCD analysis of structure functions with the help of Jacobi polynomials

The method of QCD analysis of singlet and nonsinglet structure functions, based on their Jacobi polynomial reconstruction from perturbative QCD predictions for the Mellin moments, is described. The accuracy and stability of the method are demonstrated with the help of BCDMS hydrogen data.     

Quantum fields presentation and generating functions of symplectic Schur functions and symplectic universal characters

This paper is concerned with construction of quantum fields presentation and generating functions of symplectic Schur functions and symplectic universal characters. The boson-fermion correspondence for these symmetric functions have been presented. In virtue of quantum fields, we derive a series of infinite order nonlinear integrable equations, namely, universal character hierarchy, symplectic KP hierarchy and symplectic universal character hierarchy, respectively. In addition, the solutions of these integrable systems have been discussed.     

Some new generating function formulae of the two-variable Hermite polynomials and their application in quantum optics

We explore the theoretical possibility of extending the usual squeezed state to those produced by nonlinear singlemode squeezing operators. We derive the wave functions of exp[-（ig/2）（（1-X2）1/2P ＋ P（1-X2）1/2）]｜0 in the coordinate representation. A new operator＇s disentangling formula is derived as a by-product.     

Classical orthogonal polynomials of a discrete variable continuous orthogonality relation

Based on the general theory, we consider the continous orthogonality property for classical polynomials of a discrete variable on nonuniform lattices.