Abstract: This technical note is concerned with the problem of stability analysis for time-delay systems. A new series of integral inequalities to bound a single integral term is presented by introducing some free matrices, which produces tighter bounds than some existing ones. Similarly, based on orthogonal polynomials defined in integral inner spaces, new series of multiple integral inequalities are presented as well, which include the existing double ones. To show the effectiveness of the proposed inequalities, their applications to stability analysis of systems with discrete and distributed delays are provided with numerical examples.

Abstract: There are a number of intriguing connections between Painleve equations and orthogonal polynomials, and this book is one of the first to provide an introduction to these. Researchers in integrable systems and non-linear equations will find the many explicit examples where Painleve equations appear in mathematical analysis very useful. Those interested in the asymptotic behavior of orthogonal polynomials will also find the description of Painleve transcendants and their use for local analysis near certain critical points helpful to their work. Rational solutions and special function solutions of Painleve equations are worked out in detail, with a survey of recent results and an outline of their close relationship with orthogonal polynomials. Exercises throughout the book help the reader to get to grips with the material. The author is a leading authority on orthogonal polynomials, giving this work a unique perspective on Painleve equations.

Abstract: Preliminary version of Chapter 2 in the book "Encyclopedia of Special functions: The Askey-Bateman Project, Vol. 2: Multivariate special functions", T. H. Koornwinder and J. V. Stokman (eds.), Cambridge University Press, 2021.

Abstract: The aim of this article is to provide an overview of polynomial chaos (PC) based methods for the statistical analysis of transmission lines. The underlying idea of PC is to represent stochastic line voltages and currents as expansions of predefined orthogonal polynomials. The determination of the expansion coefficients allows obtaining pertinent statistical information and is generally much faster than running, e.g., a Monte Carlo (MC) analysis. There exist several approaches to calculate the PC expansion coefficients. The article briefly reviews virtually all existing methods, whilst focusing on the popular and accurate stochastic Galerkin (SG) method as well as on the recent, more efficient and non-intrusive formulation of the so-called stochastic testing (ST) method. These two techniques are introduced by way of a simple illustrative example, i.e., a single-wire line running above a ground plane. Numerical comparisons in terms of accuracy and efficiency are also provided for a four-wire line with nonlinear terminations.

Abstract: We choose a complete set of square integrable functions as a basis for the expansion of the wavefunction in configuration space such that the matrix representation of the nonrelativistic time-independent linear wave operator is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. The recursion relation is then solved exactly in terms of orthogonal polynomials in the energy. Some of these polynomials are not found in the mathematics literature. The asymptotics of these polynomials give the phase shift for the continuous energy scattering states and the spectrum for the discrete energy bound states. Depending on the space and boundary conditions, the basis functions are written in terms of either the Laguerre or Jacobi polynomials. The tridiagonal requirement limits the number of potential functions that yield exact solutions of the wave equation. Nonetheless, the class of exactly solvable prob...

Abstract: We describe a highly efficient numerical scheme for finding two-sided bounds for the eigenvalues of the fractional Laplace operator (−Δ)α/2 in the unit ball D⊂Rd, with a Dirichlet condition in the complement of D. The standard Rayleigh–Ritz variational method is used for the upper bounds, while the lower bounds involve the lesser known Aronszajn method of intermediate problems. Both require explicit expressions for the fractional Laplace operator applied to a linearly dense set of functions in L2(D). We use appropriate Jacobi-type orthogonal polynomials, which were studied in a companion paper (B. Dyda, A. Kuznetsov and M. Kwaśnicki, ‘Fractional Laplace operator and Meijer G-function’, Constr. Approx., to appear, doi:10.1007/s00365-016-9336-4). Our numerical scheme can be applied analytically when polynomials of degree two are involved. This is used to partially resolve the conjecture of Kulczycki, which claims that the second smallest eigenvalue corresponds to an antisymmetric function: we prove that this is the case when either d⩽2 and α∈(0,2], or d⩽9 and α=1, and we provide strong numerical evidence for d⩽9 and general α∈(0,2].

Abstract: We introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line, with correlation kernels defined via spectral projections for the associated Jacobi matrices. For classical weights, we show how such ensembles arise as limits of various hypergeometric orthogonal polynomial ensembles. We then prove that the q-Laplace transform of the height function of the ASEP with step initial condition is equal to the expectation of a simple multiplicative functional on a discrete Laguerre ensemble—a member of the new family. This allows us to obtain the large time asymptotics of the ASEP in three limit regimes: (a) for finitely many rightmost particles; (b) GUE Tracy–Widom asymptotics of the height function; (c) KPZ asymptotics of the height function for the ASEP with weak asymmetry. We also give similar results for two instances of the stochastic six vertex model in a quadrant. The proofs are based on limit transitions for the corresponding determinantal point processes.

Abstract: We study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in \cite{Ristivojevic} and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasi-momenta, the ground-state energy, the local two-body correlation function and Tan's contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.

Abstract: We present a framework for the construction of linearizations for scalar and matrix polynomials based on dual bases which, in the case of orthogonal polynomials, can be described by the associated recurrence relations. The framework provides an extension of the classical linearization theory for polynomials expressed in nonmonomial bases and allows us to represent polynomials expressed in product families, that is, as a linear combination of elements of the form $\phi_i(\lambda) \psi_j(\lambda)$, where $\{ \phi_i(\lambda) \}$ and $\{ \psi_j(\lambda) \}$ can either be polynomial bases or polynomial families which satisfy some mild assumptions. We show that this general construction can be used for many different purposes. Among them, we show how to linearize sums of polynomials and rational functions expressed in different bases. As an example, this allows us to look for intersections of functions interpolated on different nodes without converting them to the same basis. We then provide some constructions ...

Abstract: In this paper, a new method based on hybridization of Lucas and Fibonacci polynomials is developed for approximate solutions of 1D and 2D nonlinear generalized BenjaminBonaMahonyBurgers equations. Firstly time discretization is made by using finite difference approaches. After that unknown function and its derivatives are expanded to Lucas series. Based on these series expansion, differentiation matrices are derived by utilizing Fibonacci polynomials. By doing so, the solution of the mentioned equations is reduced to the solution of an algebraic system of equations. By solving this system of equations the Lucas series coefficients are obtained. Then substituting these coefficients into Lucas series expansion approximate solutions can be constructed successively. The main goal of this paper is to indicate that Lucas polynomial based method is appropriate for 1D and 2D nonlinear problems. Efficiency and performance of the proposed method are judged on six test problems which consists of the 1D and 2D version of mentioned equation by calculating L2 and L error norms. Feasibility of the method is verified by obtained accurate results.

Abstract: We choose a complete set of square integrable functions as basis for the expansion of the wavefunction in configuration space such that the matrix representation of the nonrelativistic time-independent wave operator is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction in this basis. The recursion relation is then solved exactly in terms of orthogonal polynomials in the energy. Some of these polynomials are not found in the mathematics literature. The asymptotics of these polynomials give the phase shift of the continuous energy scattering states and the spectrum for the discrete energy bound states. Depending on the space and boundary conditions, the basis functions are written in terms of either the Laguerre or Jacobi polynomials. The tridiagonal requirement limits the number of potential functions that yield exact solutions of the wave equation. Nonetheless, the class of exactly solvable problems in this approach is larger than the conventional class (see Table 12). We also give very accurate results for cases where the wave operator matrix is not tridiagonal but its elements could be evaluated either exactly or numerically with high precision.

Abstract: We consider polynomials on spaces of -summing sequences of -dimensional complex vectors, which are symmetric with respect to permutations of elements of the sequences, and describe algebraic bases of algebras of continuous symmetric polynomials on

Abstract: In the recent years a significant progress was achieved in the field of design and fabrication of optical systems based on freeform optical surfaces. They provide a possibility to build fast, wide-angle and high-resolution systems, which are very compact and free of obscuration. However, the field of freeform surfaces design techniques still remains underexplored. In the present paper we use the mathematical apparatus of orthogonal polynomials defined over a square aperture, which was developed before for the tasks of wavefront reconstruction, to describe shape of a mirror surface. Two cases, namely Legendre polynomials and generalization of the Zernike polynomials on a square, are considered. The potential advantages of these polynomials sets are demonstrated on example of a three-mirror unobscured telescope with F/# = 2.5 and FoV = 7.2x7.2°. In addition, we discuss possibility of use of curved detectors in such a design.

Abstract: The aim of this paper is to give generating functions and to prove various properties for some new families of special polynomials and numbers. Several interesting properties of such families and their connections with other polynomials and numbers of the Bernoulli, Euler, Apostol–Bernoulli, Apostol–Euler, Genocchi and Fibonacci type are presented. Furthermore, the Fibonacci-type polynomials of higher order in two variables and a new family of special polynomials $$(x,y)\mapsto \mathbb {G}_{d}(x,y;k,m,n)$$ , including several particular cases, are introduced and studied. Finally, a class of polynomials and corresponding numbers, obtained by a modification of the generating function of Humbert’s polynomials, is also considered.

Abstract: In the recent years a significant progress was achieved in the field of design and fabrication of optical systems based on freeform optical surfaces. They provide a possibility to build fast, wide-angle and high-resolution systems, which are very compact and free of obscuration. However, the field of freeform surfaces design techniques still remains underexplored. In the present paper we use the mathematical apparatus of orthogonal polynomials defined over a square aperture, which was developed before for the tasks of wavefront reconstruction, to describe shape of a mirror surface. Two cases, namely Legendre polynomials and generalization of the Zernike polynomials on a square, are considered. The potential advantages of these polynomials sets are demonstrated on example of a three-mirror unobscured telescope with F/#=2.5 and FoV=7.2x7.2°. In addition, we discuss possibility of use of curved detectors in such a design.

Abstract: In this paper, we obtain initial coefficient bounds for functions belong to a subclass of bi-univalent functions by using the Chebyshev polynomials and also we find Fekete-Szegö inequalities for this class.

Abstract: We prove an explicit formula for the Poincare polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of a parabolic vector bundles and a nilpotent endomorphism of prescribed generic type. The generating function counting these pairs is shown to be a product of Macdonald polynomials and the function counting pairs without parabolic structure. The modified Macdonald polynomial $\tilde H_\lambda[X;q,t]$ is interpreted as a weighted count of points of the affine Springer fiber over the constant nilpotent matrix of type $\lambda$.

Abstract: Orthogonal polynomials (OPs) are beneficial for image processing OPs are used to reflect an image or a scene to a moment domain, and moments are subsequently used to extract object contours utilised in various applications In this study, OP-based edge detection operators are introduced to replace traditional convolution-based and block processing methods with direct matrix multiplication A mathematical model with empirical study results is established to investigate the performance of the proposed detectors compared with that of traditional algorithms, such as Sobel and Canny operators The proposed operators are then evaluated by using entire images from a well-known data set Experimental results reveal that the proposed operator achieves a more favourable interpretation, especially for images distorted by motion effects, than traditional methods do

Abstract: This paper introduces a new generalized polynomial chaos expansion (PCE) comprising multivariate Hermite orthogonal polynomials in dependent Gaussian random variables. The second-moment properties of Hermite polynomials reveal a weakly orthogonal system when obtained for a general Gaussian probability measure. Still, the exponential integrability of norm allows the Hermite polynomials to constitute a complete set and hence a basis in a Hilbert space. The completeness is vitally important for the convergence of the generalized PCE to the correct limit. The optimality of the generalized PCE and the approximation quality due to truncation are discussed. New analytical formulae are proposed to calculate the mean and variance of a generalized PCE approximation of a general output variable in terms of the expansion coefficients and statistical properties of Hermite polynomials. However, unlike in the classical PCE, calculating the coefficients of the generalized PCE requires solving a coupled system of linear equations. Besides, the variance formula of the generalized PCE contains additional terms due to statistical dependence among Gaussian variables. The additional terms vanish when the Gaussian variables are statistically independent, reverting the generalized PCE to the classical PCE. Numerical examples illustrate the generalized PCE approximation in estimating the statistical properties of various output variables.

Abstract: Recently, several authors have studied the degenerate Bernoulli and Euler polynomials and given some intersting identities of those polynomials. In this paper, we consider the degenerate Bell numbers and polynomials and derive some new identities of those numbers and polynomials associated with special numbers and polynomials. In addition, we investigate some properties of the degenerate Bell polynomials which are derived by using the notion of composita. From our investigation, we give some new relations between the degenerate Bell polynomials and the special polynomials.

Abstract: A two-dimensional/one-dimensional (2D/1D) variational nodal approach is presented for pressurized water reactor core calculations without fuel-moderator homogenization A 2D/1D approximation to the within-group neutron transport equation is derived and converted to an even-parity form The corresponding nodal functional is presented and discretized to obtain response matrix equations Within the nodes, finite elements in the x-y plane and orthogonal functions in z are used to approximate the spatial flux distribution On the radial interfaces, orthogonal polynomials are employed; on the axial interfaces, piecewise constants corresponding to the finite elements eliminate the interface homogenization that has been a challenge for method of characteristics (MOC)–based 2D/1D approximations The angular discretization utilizes an even-parity integral method within the nodes, and low-order spherical harmonics (PN) on the axial interfaces The x-y surfaces are treated with high-order PN combined with qua

Abstract: This paper presents two novel general summation inequalities, respectively, in the upper and lower discrete regions. Thanks to the orthogonal polynomials defined in different inner spaces, various concrete single/multiple summation inequalities are obtained from the two general summation inequalities, which include almost all of the existing summation inequalities, e.g., the Jensen, the Wirtinger-based and the auxiliary function-based summation inequalities. Based on the new summation inequalities, a less conservative stability condition is derived for discrete-time systems with time-varying delay. Numerical examples are given to show the effectiveness of the proposed approach.

Abstract: This chapter begins with a discussion of interpolation. Polynomial interpolation for a function of a single variable is analyzed, and implemented through Newton, Lagrange and Hermite forms. Intelligent selection of interpolation points is discussed, and extensions to multi-dimensional polynomial interpolation are presented. Rational polynomial interpolation is studied next, and connected to quadric surfaces. Then the discussion turns to piecewise polynomial interpolation and splines. The study of interpolation concludes with a presentation of parametric curves. Afterwards, the chapter moves on to least squares approximation, orthogonal polynomials and trigonometric polynomials. Trigonometric polynomial interpolation or approximation is implemented by the fast Fourier transform. The chapter concludes with wavelets, as well as their application to discrete data and continuous functions.

Abstract: The need to solve polynomial eigenvalue problems for matrix polynomials expressed in nonmonomial bases has become very important. Among the most important bases in numerical applications are the Chebyshev polynomials of the first and second kind. In this work, we introduce a new approach for constructing strong linearizations for matrix polynomials expressed in Chebyshev bases, generalizing the classical colleague pencil, and expanding the arena in which to look for linearizations of matrix polynomials expressed in Chebyshev bases. We show that any of these linearizations is a strong linearization regardless of whether the matrix polynomial is regular or singular. In addition, we show how to recover eigenvectors, minimal indices, and minimal bases of the polynomial from those of any of the new linearizations. As an example, we also construct strong linearizations for matrix polynomials of odd degree that are symmetric (resp., Hermitian) whenever the matrix polynomials are symmetric (resp., Hermitian).