The peculiar (monic) polynomials,the zeros of which equal their coefficients

We evaluate the number of complex monic polynomials, of arbitrary degree N, the zeros of which are equal to their coefficients. In the following, we call polynomials with this property peculiar polynomials. We further show that the problem of determining the peculiar polynomials of degree N simplifies when any of the coefficients is either 0 or 1. We proceed to estimate the numbers of peculiar polynomials of degree N having one coefficient zero, or one coefficient equal to one, or neither.     

Deconvolution of 3-D Gaussian kernels

Ulmer and Kaissl formulas for the deconvolution of one-dimensional Gaussian kernels are generalized to the three-dimensional case. The generalization is based on the use of the scalar version of the Grad's multivariate Hermite polynomials which can be expressed through ordinary Hermite polynomials.     

The Origin and Mathematical Characteristics of the Super-Universal Associated-Legendre Polynomials

We study the mathematical characteristics of the super-universal associated-Legendre polynomials arising from a kind of double ring-shaped potentials and obtain their polar angular wave functions with certain parity. We find that there exists the even or odd parity for the polar angular wave functions when the parameter η is taken to be positive integer, while there exist both even and odd parities when η is taken as positive non-integer real values. The relations among the super-universal associated-Legendre polynomials, the hypergeometric polynomials, and the Jacobi polynomials are also established.     

The PDEs of Biorthogonal Polynomials Arising in the Two-Matrix Model

The two-matrix model can be solved by introducing biorthogonal polynomials. In the case the potentials in the measure are polynomials, finite sequences of biorthogonal polynomials (called windows) satisfy polynomial ODEs as well as deformation equations (PDEs) and finite difference equations (ΔE) which are all Frobenius compatible and define discrete and continuous isomonodromic deformations for the irregular ODE, as shown in previous works of ours. In the one matrix model an explicit and concise expression for the coefficients of these systems is known and it allows to relate the partition function with the isomonodromic tau-function of the overdetermined system. Here, we provide the generalization of those expressions to the case of biorthogonal polynomials, which enables us to compute the determinant of the fundamental solution of the overdetermined system of ODE + PDEs + ΔE.     

环扇形分块主镜的波面拟合及重构方法研究

用有限元分析获得了环扇形分块主镜的镜面热变形数据,分别用圆域Zernike多项式和环扇域Zernike多项式对其进行了波面拟合,比较了用两种多项式的拟合精度。基于哈特曼-夏克波前传感器对由环扇形镜面热变形引起的畸变波前进行了模式法重构,分析了用上述两种多项式重构时矩阵条件数和测量误差对重构精度的影响。     

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.     

New Properties of the Zeros of Krall Polynomials

We identify a new class of algebraic relations satisfied by the zeros of orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two, known as Krall polynomials. Given an orthogonal polynomial family , we relate the zeros of the polynomial p_N with the zeros of p_m for each m ≤ N (the case m = N corresponding to the relations that involve the zeros of p_N only). These identities are obtained by finding exact expressions for the similarity transformation that relates the spectral and the (interpolatory) pseudospectral matrix representations of linear differential operators, while using the zeros of the polynomial p_N as the interpolation nodes. The proposed framework generalizes known properties of classical orthogonal polynomials to the case of nonclassical polynomial families of Krall type. We illustrate the general result by proving new identities satisfied by the Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi orthogonal polynomials.     

Classical Skew Orthogonal Polynomials and Random Matrices

Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the case that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed-form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre, and Jacobi cases.     

厄米多项式的定义及其他

分析了关于厄米多项式的争论.介绍了厄米多项式的两种定义.     

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 the Zeros of Polynomials Satisfying Certain Linear Second-Order ODEs Featuring Many Free Parameters

Certain techniques to obtain properties of the zeros of polynomials satisfying second-order ODEs are reviewed. The application of these techniques to the classical polynomials yields formulas which were already known; new are instead the formulas for the zeros of the (recently identified, and rather explicitly known) polynomials satisfying a (recently identified) second-order ODE which features many free parameters and only polynomial solutions. Some of these formulas have a Diophantine connotation. Techniques to manufacture infinite sequences of second-order ODEs featuring only polynomial solutions are also reported.     

An Integrable Chain and Bi-Orthogonal Polynomials

An integrable chain connected to the isospectral evolution of the polynomials of type R–I introduced by Ismail and Masson is presented. The equations of motion of this chain generalize the corresponding equations of the relativistic Toda chain introduced by Ruijsenaars. We study simple self-similar solutions to these equations that are obtained through separation of variables. The corresponding polynomials are expressed in terms of the Gauss hypergeometric function. It is shown that these polynomials are stable (up to shifts of the parameters) against Darboux transformations of the generalized chain.     

On Integrals Involving Universal Associated Legendre Polynomials and Powers of the Factor (1-x2) and Their Byproducts

The associated Legendre polynomials play an important role in the central fields, but in the case of the non-central field we have to introduce the universal associated Legendre polynomials P_l'^m'(x) when studying the modified Pschl-Teller potential and the single ring-shaped potential. We present the evaluations of the integrals involving the universal associated Legendre polynomials and the factor (1-x²)^-p-1 as well as some important byproducts of this integral which are useful in deriving the matrix elements in spin-orbit interaction. The calculations are obtained systematically using some properties of the generalized hypergeometric series.     

Non-Hermitian oscillator Hamiltonians and multiple Charlier polynomials

A set of r non-Hermitian oscillator Hamiltonians in r dimensions is shown to be simultaneously diagonalizable. Their spectra are real and the common eigenstates are expressed in terms of multiple Charlier polynomials. An algebraic interpretation of these polynomials is thus achieved and the model is used to derive some of their properties.     

自适应光学系统的模式法数值模拟

建立了利用模式法笃自适应光学系统进行数值模拟的理论模型，编制了计算程序，并与激光大气传输计算程序衔接起来，进行了大量数值模拟计算。首次发现：存在泽尼特多项式展开的最佳项数。大于一定项数的展开式的效果迅速变坏，竖排和斜排经特面式展开有类似的结果。文献中认为可以采用的１５项经特多项式展开的效果不好，最佳项数随着横向风速的增加而减小，在风速较大时最佳项数下的模范地结果稍好于直接斜率控制法的结果。     

Applications of Commuting Difference Operators to Orthogonal Polynomials in Several Variables

Elementary properties of the Koornwinder--Macdonald multivariable Askey--Wilson polynomials are discussed. Studied are the orthogonality, the difference equations, the recurrence relations, and the orthonormalization constants for these polynomials. Essential in our approach are certain commuting difference operators simultaneously diagonalized by the polynomials.     

Romanovski polynomials in selected physics problems

We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality properties. We then focus on the family of polynomials which exhibits a finite orthogonality. This family, to be referred to as the Romanovski polynomials, is required in exact solutions of several physics problems ranging from quantum mechanics and quark physics to random matrix theory. It appears timely to draw attention to it by the present study. Our survey also includes several new observations on the orthogonality properties of the Romanovski polynomials and new developments from their Rodrigues formula.     

Automorphisms of the q-deformed algebra suq(1, 1) and d-Orthogonal polynomials of q-Meixner type

Starting from an operator given as a product of q-exponential functions in irreducible representations of the positive discrete series of the q-deformed algebra su_q(1, 1), we express the associated matrix elements in terms of d-orthogonal polynomials. An algebraic setting allows to establish some properties : recurrence relation, generating function, lowering operator, explicit expression and d-orthogonality relations of the involved polynomials which are reduced to the orthogonal q-Meixner polynomials when d=1. If q ↑ 1, these polynomials tend to some d-orthogonal polynomials of Meixner type.     

Modal Description of Wavefront Aberration in Non-circle Apertures

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