Tomographic probability and entropic inequalities for special functions

We discussed some aspects of the tomographic-probability representation of quantum mechanics. Using known generic inequalities for Shannon and relative entropies, we obtain some new inequalities for special functions such as Laguerre, Legendre, and two-variable Hermite polynomials.     

Entropy and information characteristics of qubit states

Shannon entropy, Rényi entropy, and Tsallis entropy are discussed for the tomographic probability distributions of qubit states. Relative entropy and its properties are considered for the tomographic probability distribution describing the states of multi-spin systems. New inequalities for Hermite polynomials are obtained.     

Kinetic theory of field effects in nonuniform molecular gases

Effects of magnetic and electric fields on transport phenomena in dilute polyatomic gases are reviewed within the framework of first order Enskog theory. The established technique of approximate operator inversion is used to give first order approximations of the transport coefficients. Instead of the customary expansion of polarization into orthogonal polynomials a more general treatment is chosen here so as to accomodate recent experimental observations. The polarizations produced by macroscopic fluxes are assumed to be eigenfunctions of the collision operator within the subspace of functions anisotropic in angular momentum. The formalism is extended to mixtures in a way to let the final expressions assume the same form as for pure gases. The obtained transport coefficients obey several symmetry relations and inequalities. Additional inequalities are now also derived for the matrix describing the saturated field effects.     

Statistical Properties of the Generalized Photon-Added Pair Coherent State

We present a class of generalized photon-added pair coherent states (GPAPCS) and analyze some prominent nonclassical properties such as sub-Poissonian distribution and violations of Cauchy-Schwarz inequalities. In addition, we derive that the Wigner function of GPAPCS involves correlation of two two-variable Hermite polynomials and its Husimi function is related to a two-variable Hermite polynomial. Their behaviors varying with the phase space parameters are also graphically discussed. We find that the nonclassical effects of GPAPCS exhibits more with increasing of excitation photon numbers.     

Correlation inequalities and the mass gap inP(φ)2

For theP()₂ field theory, we prove that the falloff of the (vacuum subtracted) two point Schwinger function dominates the higher order (vacuum subtracted) Schwinger functions. As applications, we prove that for even polynomials, the first excited state is odd, and that when there is a one particle state in the infinite volume limit, it is coupled to the vacuum by a single power of the field. The main inputs are the theory of Markov fields and the F.K.G. inequalities.     

An Approximation Scheme for Constructing π°π° Amplitudes from ACU Requirements

The requirements of analyticity, crossing symmetry and unitarity (ACU) are used as input to construct amplitudes for the scattering of neutral π mesons. They are obtained explicitly as polynomials in certain conformally transformed variables which ensure the correct analyticity structure. Crossing symmetry is expressed as a set of linear relations between the expansion coefficients. The same is true for the threshold behaviour of the partial waves and the behaviour for large energies of either the partial waves and the amplitudes itself. To enforce the positivity of the partial waves (up to a certain l) a new set of variables is introduced. It is obtained by solving the system of linear inequalities expressing the positivity condition. In this way one gets explicit expressions for all amplitudes satisfying the ACU requirements to some approximation. The rigorous constraints obtained from these ACU requirements are used to test the accuracy of this construction. The properties of all amplitudes are investigated in detail. It is indicated how this approximation scheme may be improved to any degree of accuracy and how it may be generalized to the full isospin case.     

基于泽尼克多项式进行面形误差拟合的频域分析

获得泽尼克多项式的频谱信息是正确利用该多项式进行误差拟合的关键。推导出了泽尼克多项式的傅里叶变换公式,在频域中分析了不同阶数该多项式的径向频谱信息和幅角频谱信息,得到了有限项泽尼克多项式能够有效表达面形误差的最大径向空间频率和角频率。基于频域分析理论,利用泽尼克多项式对不同口径局部误差进行了拟合,并利用齐戈（Zygo）干涉仪对带有不同面形误差的光学元件进行了试验分析。结果表明,当误差的径向空间频率或角频率超出泽尼克多项式所能表达的频谱范围时,拟合误差迅速变大。     

On λ-Bell polynomials associated with umbral calculus

In this paper, we introduce some new λ-Bell polynomials and Bell polynomials of the second kind and investigate properties of these polynomials. Using our investigation, we derive some new identities for the two kinds of λ-Bell polynomials arising from umbral calculus.     

Modal Description of Wavefront Aberration in Non-circle Apertures

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Integral representations for the Lagrange polynomials,Shively’s pseudo-Laguerre polynomials,and the generalized Bessel polynomials

Motivated essentially by their potential for applications in the mathematical, physical, and statistical sciences, the object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing the main results presented here, the corresponding integral representations are derived for familiar simpler classes of hypergeometric polynomials such as (for example) the Lagrange polynomials, Shively’s pseudo-Laguerre polynomials, and generalized Bessel polynomials. Each of the integral representations derived in this paper may be also viewed as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.     

Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials, b) those constructed from Cauchy transforms of the same orthogonal polynomials, and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the Riemann-Hilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via the Deift-Zhou steepest-descent/stationary phase method for Riemann-Hilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for an arbitrary invariant ensemble of =2 symmetry class.     

基于环扇域正交多项式的频域分析

利用傅里叶变换得到了Zernike多项式和环扇域内正交多项式的功率谱密度（PSD）分布,以及正交多项式每项所对应的峰值径向空间频率和半峰值径向空间频率范围。通过对比发现,正交多项式与相同阶的Zernike多项式PSD分布相似,但是却含有更高的空间频率成分。通过计算机仿真,发现正交多项式中每一项都基本上只代表特定的空间频率范围,根据相位度量的环扇形镜面面形空间频率分布,选择适当的正交多项式的项进行拟合,不仅能够节省运算时间,而且还可以保证拟合精度。     

Some generalized Lagrange-based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

In this paper, we introduce a general family of Lagrange-based Apostol-type polynomials thereby unifying the Lagrange-based Apostol-Bernoulli and the Lagrange-based Apostol-Genocchi polynomials. We also define Lagrange-based Apostol-Euler polynomials via the generating function. In terms of these generalizations, we find new and useful relations between the unified family and the Apostol-Euler polynomials. We also derive their explicit representations and list some basic properties of each of them. Further relations between the above-mentioned polynomials, including a family of bilinear and bilateral generating functions, are given. Moreover, a generating relation involving the Stirling numbers of the second kind is derived.     

Paraxial and nonparaxial polynomial beams and the analytic approach to propagation

We construct solutions of the paraxial and Helmholtz equations that are polynomials in their spatial variables. These are derived explicitly by using the angular spectrum method and generating functions. Paraxial polynomials have the form of homogeneous Hermite and Laguerre polynomials in Cartesian and cylindrical coordinates, respectively, analogous to heat polynomials for the diffusion equation. Nonparaxial polynomials are found by substituting monomials in the propagation variable z with reverse Bessel polynomials. These explicit analytic forms give insight into the mathematical structure of paraxially and nonparaxially propagating beams, especially in regard to the divergence of nonparaxial analogs to familiar paraxial beams.     

Special polynomials associated with some hierarchies

Special polynomials associated with rational solutions of a hierarchy of equations of Painlevé type are introduced. The hierarchy arises by similarity reduction from the Fordy-Gibbons hierarchy of partial differential equations. Some relations for these special polynomials are given. Differential-difference hierarchies for finding special polynomials are presented. These formulae allow us to obtain special polynomials associated with the hierarchy studied. It is shown that rational solutions of members of the Schwarz-Sawada-Kotera, the Schwarz-Kaup-Kupershmidt, the Fordy-Gibbons, the Sawada-Kotera and the Kaup-Kupershmidt hierarchies can be expressed through special polynomials of the hierarchy studied.     

A note on <Emphasis Type="Italic">q</Emphasis>-Bernstein polynomials

Recently, Simsek-Acikgoz [17] and Kim-Jang-Yi [9] have studied the q-extension of Bernstein polynomials. In the present paper, we suggest q-extensions of Bernstein polynomials of degree n which differ from the q-Bernstein polynomials of Simsek-Acikgoz [17] and Kim-Jang-Yi [9]. Using these q-Bernstein polynomials, we derive fermionic p-adic integral representations of several q-Bernstein-type polynomials. Finally, we investigate identities between q-Bernstein polynomials and q-Euler numbers.     

Representation of the quantum mechanical wavefunction by orthogonal polynomials in the energy and physical parameters

We present a formulation of quantum mechanics based on the theory of orthogonal polynomials.The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. Information about the corresponding physical systems(both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable quantum mechanical problems becomes larger than in the conventional formulation(see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases(the Meixner–Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials.     

Orthonormal polynomials in analysis of single-mode fiber coupling

Zernike polynomials have been widely used for wave-front analysis because of their orthogonality over a uniform circular pupil. However, the pupil is not uniform but weighted by the backpropagated fiber mode in analyzing fiber coupling efficiency. Zernike polynomials are not appropriate for a weighted pupil due to their lack of orthogonality over such pupil. We emphasize the advantages of using orthonormal polynomials in fiber coupling systems. The orthonormal polynomials over weighted pupil are derived by matrix approach. The effects of primary aberrations are investigated based on the orthonormal polynomials. The accuracy of the Strehl ratio approximation for primary aberrations is evaluated.     

The generalized Yablonskii-Vorob'ev polynomials and their properties

Rational solutions of the generalized second Painlevé hierarchy are classified. Representation of the rational solutions in terms of special polynomials, the generalized Yablonskii-Vorob'ev polynomials, is introduced. Differential-difference relations satisfied by the polynomials are found. Hierarchies of differential equations related to the generalized second Painlevé hierarchy are derived. One of these hierarchies is a sequence of differential equations satisfied by the generalized Yablonskii-Vorob'ev polynomials.