Gegenbauer Orthogonal Polynomials Research Articles

The symmetric Dunkl-classical orthogonal polynomials revisited

ABSTRACT We investigate the symmetric Dunkl-classical orthogonal polynomials by using a new approach in connection with the Dunkl operator. The main aim of this technique is first and foremost to determine the recurrence coefficients. We establish the existence and uniqueness of symmetric Dunkl-classical orthogonal polynomials, and confirm that the two families of generalized Hermite orthogonal polynomials and Gegenbauer orthogonal polynomials are the only ones belonging to this class. Two apparently new characterizations in a more general setting are given. This paper complements earlier work of Ben Cheikh and Gaied [Characterization of the Dunkl-classical symmetric orthogonal polynomials. Appl Math Comput. 2007;187:105–114. doi: 10.1016/j.amc.2006.08.108].

Monotone complexity measures of multidimensional quantum systems with central potentials

In this work, we explore the (inequality-type) properties of the monotone complexity-like measures of the internal complexity (disorder) of multidimensional non-relativistic electron systems subject to a central potential. Each measure quantifies the combined balance of two spreading facets of the electron density of the system. We show that the hyperspherical symmetry (i.e., the multidimensional spherical symmetry) of the potential allows Cramér–Rao, Fisher–Shannon, and Lopez-Ruiz, Mancini, Calbet–Rényi complexity measures to be expressed in terms of the space dimensionality and the hyperangular quantum numbers of the electron state. Upper bounds, mutual complexity relationships, and complexity-based uncertainty relations of position–momentum type are also found by means of the electronic hyperangular quantum numbers and, at times, the Heisenberg–Kennard relation. We use a methodology that includes a variational approach with a covariance matrix constraint and some algebraic linearization techniques of hyperspherical harmonics and Gegenbauer orthogonal polynomials.

Applications of Gegenbauer Polynomials for Subfamilies of Bi-Univalent Functions Involving a Borel Distribution-Type Mittag-Leffler Function

In this research, a novel linear operator involving the Borel distribution and Mittag-Leffler functions is introduced using Hadamard products or convolutions. This operator is utilized to develop new subfamilies of bi-univalent functions via the principle of subordination with Gegenbauer orthogonal polynomials. The investigation also focuses on the estimation of the coefficients |aℓ|(ℓ=2,3) and the Fekete–Szegö inequality for functions belonging to these subfamilies of bi-univalent functions. Several corollaries and implications of the findings are discussed. Overall, this study presents a new approach for constructing bi-univalent functions and provides valuable insights for further research in this area.

Threshold condensation to singular support for a Riesz equilibrium problem

We compute the equilibrium measure in dimension $$d=s+4$$ associated to a Riesz s-kernel interaction with an external field given by a power of the Euclidean norm. Our study reveals that the equilibrium measure can be a mixture of a continuous part and a singular part. Depending on the value of the power, a threshold phenomenon occurs and consists of a dimension reduction or condensation on the singular part. In particular, in the logarithmic case $$s=0$$ ( $$d=4$$ ), there is condensation on a sphere of special radius when the power of the external field becomes quadratic. This contrasts with the case $$d=s+3$$ studied previously, which showed that the equilibrium measure is fully dimensional and supported on a ball. Our approach makes use, among other tools, of the Frostman or Euler–Lagrange variational characterization, the Funk–Hecke formula, the Gegenbauer orthogonal polynomials, and hypergeometric special functions.

Parameters Estimation in Non-Negative Integer-Valued Time Series: Approach Based on Probability Generating Functions

This manuscript deals with a parameter estimation of a non-negative integer-valued (NNIV) time series based on the so-called probability generating function (PGF) method. The theoretical background of the PGF estimation technique for a very general, stationary class of NNIV time series is described, as well as the asymptotic properties of the obtained estimates. After that, a particular emphasis is given to PGF estimators of independent identical distributed (IID) and integer-valued non-negative autoregressive (INAR) series. A Monte Carlo study of the thus obtained PGF estimates, based on a numerical integration of the appropriate objective function, is also presented. For this purpose, numerical quadrature formulas were computed using Gegenbauer orthogonal polynomials. Finally, the application of the PGF estimators in the dynamic analysis of some actual data is given.

A Novel Gegenbauer Wavelet-Based Approach for Stability and Surface Location Error Analyses of Milling Process

Abstract This paper proposes a novel wavelet-based method for the computation of the milling stability and surface location error (SLE). The governing equation of milling processes considering the self-vibration and forced-vibration is formulated as a delayed differential equation with state space form. The total time period is considered as a whole, sampled by the Chebyshev nodes. Then, the vibration displacement corresponding to each sampling grid point is fitted by a weighted linear sum of Gegenbauer orthogonal polynomials. A new method named direct differentiation based algorithm (DDBA) is proposed to determine these weight coefficients, which gets rid of dependence on the classic differential operator. On the one hand, the Floquet matrix is constructed to calculate the milling stability. On the other hand, the coefficient matrix independent of time is established to predict the SLE. The benchmark and experimentally verified examples are employed to validate the effectiveness of the proposed approach. The optimal combination of orthogonal polynomials is fixed by exploring the effect of the control parameter on the predictive accuracy. At last, the calculation vulnerability that can cause the prediction accuracy to be dropped obviously in some step lengths is revealed and repaired.

Hertzian and adhesive plane models of contact of two inhomogeneous elastic bodies

Previous study of contact of power-law graded materials concerned the contact of a rigid body (punch) with an elastic inhomogeneous foundation whose inhomogeneity is characterised by the Young modulus varying with depth as a power function. This paper models Hertzian and adhesive contact of two elastic inhomogeneous power-law graded bodies with different exponents. The problem is governed by an integral equation with two different power kernels. A nonstandard method of Gegenbauer orthogonal polynomials for its solution is proposed. It leads to an infinite system of linear algebraic equations of a special structure. The integral representations of the system coefficients are evaluated, and the properties of the system are studied. It is shown that if the exponents coincide, the infinite system admits a simple exact solution that corresponds to the case when the Young moduli are different but the exponents are the same. Formulas for the length of the contact zone, the pressure distribution and the surface normal displacements of the contacting bodies are obtained in the form convenient for computations. Effects of the mismatch in the Young moduli exponents are studied. A comparative analysis of the Hertzian and adhesive contact models clarifies the effects of the surface energy density on the contact pressure, the contact zone size and the profile of the contacting bodies outside the contact area.

Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane

We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polynomials C_n^{(1+alpha )}(z) for alpha >-1 containing the Legendre polynomials P_n(z) and the subset P_n^{(alpha +frac{1}{2},pm frac{1}{2})}(z) of the Jacobi polynomials. These polynomials provide an orthonormal basis and the corresponding weighted Bergman space forms a complete metric space. This leads to a certain family of Selberg integrals in the complex plane. We recover the known orthogonality of Chebyshev polynomials of the first up to fourth kind. The limit alpha rightarrow infty leads back to the known Hermite polynomials orthogonal in the entire complex plane. When the ellipse degenerates to a circle we obtain the weight function and monomials known from the determinantal point process of the ensemble of truncated unitary random matrices.

Indefinite integrals of special functions from hybrid equations

ABSTRACTElementary linear first and second order differential equations can always be constructed for twice differentiable functions by explicitly including the function's derivatives in the definition of these equations. If the function also obeys a conventional differential equation, information from this equation can be introduced into the elementary equations to give blended linear equations which are here called hybrid equations. Integration theorems are derived for these hybrid equations and several universal integrals are also derived. The paper presents integrals derived with these methods for cylinder functions, associated Legendre functions, and the Gegenbauer, Chebyshev, Hermite, Jacobi and Laguerre orthogonal polynomials. All the results presented have been checked using Mathematica.

Gradient Path Labelling method and tracking method for calculation of solar differential rotation using coronal bright points

Abstract With new space missions, such as Solar Dynamics Observatory (SDO), solar images are being produced in unprecedented volumes. To derive, as much as possible, information on evolution of solar activity from those huge datasets, the scientific community needs a new generation of software tools for automatic and efficient data processing. In the last decade, several research teams have been developing tools for obtaining more precise estimations of the solar rotation profile, but more are needed to improve knowledge about solar activity. We applied here a segmentation algorithm called Gradient Path Labelling (GPL), used originally to identify drusens in medical retinal images, to detect and track the coronal bright points (CBPs) using images from the AIA instrument onboard the SDO satellite. The CBPs have a tendency to change shape and size along time, to disappear and reappear at a corresponding heliographic position, therefore, decision trees were also included in the tracking solution. Since our CBP detection algorithm uses an active region mask to filter out the CBPs, whose centroid is inside the active regions, the number of identifications clearly depends on the level of solar activity. Our approach uses the commonly applied fitting relation to the latitudinal dependence of the rotational velocity, which resulted in calculation of the optimum fit parameters as well as the Gegenbauer orthogonal polynomials. Comparison of these parameters with the results presented in recent papers on this topic shows that our rotational velocity profile indicates slightly lower rotational velocities than the profiles obtained with other approaches. We also calculate the meridional motion of the CBPs, but comparison with other authors results, clearly show that a 3-day time interval is too short to estimate the latitudinal dependence of the CBP meridional motion. Distributions of the rotational velocity and meridional motion velocity uncertainties show that 85% of uncertainty values are lower than 1 degree/day. The evaluation of our test results shows that the applied algorithm is a promising tool that can help to refine the solar rotational profile.

Numerical simulation of random fields with a high-order polynomial based Ritz–Galerkin approach

The uncertainty simulation related to a spatially varying physical property requires the theory of the random field discretization. A random field in definition consists of an infinite number of random variables that are attached to all points within the simulation domain. The utility of the Karhunen–Loève (K–L) expansion allows representing a second-order random field as the summation of a few deterministic functions and standard random variables. However, the K–L expansion depends on exact eigensolutions of the Fredholm integral equation of the second kind. Along with an emerging trend of the random field simulation combined with a complex geometry, analytical expansion results are increasingly inapplicable in engineering reality. To this end, the paper presents a high-order polynomial based Ritz–Galerkin approach. To implement, the Legendre, the Chebyshev, and the Gegenbauer orthogonal polynomials are used to realize the Ritz–Galerkin approximation. Together with the Gaussian-quadrature based interpolation scheme in approximating covariance models, numerical results for eigensolutions have demonstrated the spectral accuracy of the proposed approach. Results for the non-stationary, non-Gaussian and multivariate random field simulations have validated the high-order polynomial based Ritz–Galerkin approach has engineering applications.

Entropy and complexity analysis of hydrogenic Rydberg atoms

The internal disorder of hydrogenic Rydberg atoms as contained in their position and momentum probability densities is examined by means of the following information-theoretic spreading quantities: the radial and logarithmic expectation values, the Shannon entropy, and the Fisher information. As well, the complexity measures of Crámer-Rao, Fisher-Shannon, and López Ruiz-Mancini-Calvet types are investigated in both reciprocal spaces. The leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the Laguerre and Gegenbauer orthogonal polynomials which control the wavefunctions of the Rydberg states in both position and momentum spaces. The associated generalized Heisenberg-like, logarithmic and entropic uncertainty relations are also given. Finally, application to linear (l = 0), circular (l = n − 1), and quasicircular (l = n − 2) states is explicitly done.

New class of filter functions generated most directly by Christoffel-Darboux formula for Gegenbauer orthogonal polynomials

A new original formulation of all pole low-pass filter functions is proposed in this article. The starting point in solving the approximation problem is a direct application of the Christoffel-Darboux formula for the set of orthogonal polynomials, including Gegenbauer orthogonal polynomials in the finite interval [−1, +1] with the application of a weighting function with a single free parameter. A general solution for the filter functions is obtained in a compact explicit form, which is shown to enable generation of the Gegenbauer filter functions in a simple way by choosing the value of the free parameter. Moreover, the proposed solution with the same criterion of approximation could be used to generate Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even (10th) and odd (11th) order are illustrated. The approximation is shown to yield a good compromise solution with respect to the filter frequency characteristics (magnitude as well as phase characteristics). The influence of tolerance of the filter critical component (inductor) on the proposed magnitude and group delay characteristics of a resistively terminated LC lossless ladder filter is analysed as well. The proposed filter functions are superior in terms of the excellent magnitude characteristic, which approximates an ideal filter almost perfectly over the entire pass-band range and exhibits the summed sensitivity function better than that of a Butterworth filter. In the article, we present the filter function solution that exhibits optimum amplitude as well as optimum group delay characteristics that are of crucial importance for implementation of digital processing as well as RF analogue parts of communication networks. Derivation of the other band range filter functions, which could be realised either by continuous or digital filters, is also generally possible with the procedure proposed in this article.

Reliability analysis of double random Duffing system

By using of Gegenbauer orthogonal polynomials，an effective numerical method for the reliability analysis of double random Duffing system with large coefficient of variation is presented. On the basis of the first passage model，the fourth-moment technique，the Edgeworth series approximation theory and the orthogonal polynomial expansion method are employed to develop a reliability analysis method systematically. The dynamic reliability and mean reliability of the system is discussed. The results obtained by this method are verified by Monte-Carlo simulations.

Asymptotic normality of the coefficients of polynomials associated with the Gegenbauer ones

A suitable transformation on the classical Gegenbauer orthogonal polynomials leads to polynomials with nonnegative coefficients. Asymptotic normality of those coefficients is established and rate of convergence and asymptotic formulas are provided. As examples the number of matching sets of paths and polygons, respectively, are approximated.

Generalized Gegenbauer orthogonal polynomials

In this paper we explore a specific semi-classical orthogonal sequence, namely the generalized Gegenbauer orthogonal polynomials (GG) which appear in many applications such as the weighted L p mean convergence of Hermite–Fejér interpolation or the chain of harmonic oscillators in the absence of externally applied forces. First we trace back the genesis of GG underlining its links with the Jacobi orthogonal polynomials. Second we establish a differential–difference relation and the second-order differential equation satisfied by this sequence. We end by giving the fourth-order differential equation satisfied by the association (of arbitrary order) of the GG.

A combinatorial formula of Leibniz type with application to Gegenbauer’s polynomials

We establish a combinatorial formula of Leibniz type, which is an identity for a certain differential polynomial. The formula leads to new quadratic relations between Gegenbauer's orthogonal polynomials.

A two-parameter family of weights for nonrecursive digital filters and antennas

We derive analytically a two-parameter family of weights for use in finite duration nonrecursive digital filters and in finite aperture antennas. This family of weights is based on the Gegenbauer orthogonal polynomials, which are a generalization of both Legendre and Chebyshev polynomials. It is shown that one parameter controls the main lobewidth and the other parameter controls the sidelobe taper. For a fixed main lobewidth, it is observed that the Gegenbauer weights can achieve a dramatic decrease in sidelobes far removed from the main lobe in exchange for a small increase in the first sidelobe adjacent to the main lobe. The Gegenbauer weights are derived first for discretely sampled apertures and filters. An appropriate limit is then taken to produce the Gegenbauer weighting function for continuously sampled apertures and filters. The continuous Gegenbauer weighting function contains the Kaiser-Bessel function as a special case. It is thus established that the Kaiser-Bessel function is implicitly based on Chebyshev polynomials of the second kind. Furthermore, the Dolph-Chebyshev/van der Maas weights are a limiting case of the discrete/continuous Gegenbauer weights.