Jacobi Polynomials Research Articles

Dynamic Analysis of Functionally Graded Porous (FGP) Elliptic Cylindrical Shell Based on Jacobi Polynomials Ritz Method

The lightweight of structure is widely applied in industrial applications, and the conflict between both dynamic stability and structural lightweight is still prominent. In this paper, functionally graded porous (FGP) elliptic cylindrical shells and panels with general boundary conditions are analyzed to explore the effect of the FGP on dynamic performance. First, the FGP elliptic cylindrical shell and panel models are established. Therein, three kinds of porosity distribution are considered, including nonsymmetric, symmetric, and uniform distributions. The energy expressions of the FGP elliptic cylindrical shell and panel are established by the first-order shear deformation theory (FSDT). To simulate various boundary conditions, the artificial spring boundary technique is employed in this study. Then, the Jacobi orthogonal polynomials and Fourier series are adopted to express the admissible displacements. Finally, the accuracy of this model is verified by comparing it with open literature and ABAQUS software. Results show that the variations of the boundary conditions, linear springs, thickness ratio, and porosity have close relation with the dynamic performance of the structure by affecting the stiffness of the structure.

Fractal Divergences of Generalized Jacobi Polynomials

The notion of entropy (including macro state entropy and information entropy) is used, among others, to define the fractal dimension. Rényi entropy constitutes the basis for the generalized correlation dimension of multifractals. A motivation for the study of the information measures of orthogonal polynomials is because these polynomials appear in the densities of many quantum mechanical systems with shape-invariant potentials (e.g., the harmonic oscillator and the hydrogenic systems). With the help of a sequence of some generalized Jacobi polynomials, we define a sequence of discrete probability distributions. We introduce fractal Kullback–Leibler divergence, fractal Tsallis divergence, and fractal Rényi divergence between every element of the sequence of probability distributions introduced above and the element of the equiprobability distribution corresponding to the same index. Practically, we obtain three sequences of fractal divergences and show that the first two are convergent and the last is divergent.

Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation

We study the sequence of monic polynomials {Sn}n⩾0, orthogonal with respect to the Jacobi-Sobolev inner product ⟨f,g⟩s=∫−11f(x)g(x)dμα,β(x)+∑j=1N∑k=0djλj,kf(k)(cj)g(k)(cj), where N,dj∈Z+, λj,k⩾0, dμα,β(x)=(1−x)α(1+x)βdx, α,β>−1, and cj∈R∖(−1,1). A connection formula that relates the Sobolev polynomials Sn with the Jacobi polynomials is provided, as well as the ladder differential operators for the sequence {Sn}n⩾0 and a second-order differential equation with a polynomial coefficient that they satisfied. We give sufficient conditions under which the zeros of a wide class of Jacobi-Sobolev polynomials can be interpreted as the solution of an electrostatic equilibrium problem of n unit charges moving in the presence of a logarithmic potential. Several examples are presented to illustrate this interpretation.

Freezing limits for Calogero–Moser–Sutherland particle models

AbstractOne‐dimensional interacting particle models of Calogero–Moser–Sutherland type with N particles can be regarded as diffusion processes on suitable subsets of like Weyl chambers and alcoves with second‐order differential operators as generators of the transition semigroups, where these operators are singular on the boundaries of the state spaces. The most relevant examples are multivariate Bessel processes and Heckman–Opdam processes in a compact and noncompact setting where in all cases, these processes are related to special functions associated with root systems. More precisely, the transition probabilities can be described with the aid of multivariate Bessel functions, Jack and Heckman–Opdam Jacobi polynomials, and Heckman–Opdam hypergeometric functions, respectively. These models, in particular, form dynamic eigenvalue evolutions of the classical random matrix models like β‐Hermite, β‐ Laguerre, and β‐Jacobi, that is, MANOVA, ensembles. In particular, Dyson's Brownian motions and multivariate Jacobi processes are included. In all cases, the processes depend on so‐called coupling parameters. We review several freezing limit theorems for these diffusions where, for fixed N, one or several of the coupling parameters tend to ∞. In many cases, the limits will be N‐dimensional normal distributions and, in the process case, Gauss processes. However, in some cases, normal distributions on half spaces and distributions related to some other ensembles appear as limits. In all cases, the limits are connected with the zeros of the classical one‐dimensional orthogonal polynomials of order N.

Quantum polynomials from deformed quantum algebras: Probability distributions, generating functions and difference equations

In this paper, we provide a novel generalization of quantum orthogonal polynomials from [Formula: see text]-deformed quantum algebras introduced in earlier works. We construct related quantum Jacobi polynomials and their probability distribution, factorial moments, recurrence relation, and governing difference equation. Surprisingly, these polynomials obey non-conventional recurrence relations. Particular cases of generalized quantum little Legendre, little Laguerre, Laguerre, Bessel, Rogers–Szegö, Stieltjes–Wigert and Kemp binomial polynomials are derived and discussed.

Simple, efficient, and accurate analysis of the flanged parallel-plate waveguide

The flanged parallel-plate waveguide is analysed based on the method of Kobayashi potential (KP) using Fourier function space. The presentation of the method is free from intricate mathematics. Standard integral identities are used for problem formulation, without direct use of Weber-Schafheitlin (WS) integrals. The Fourier function space is exploited for the construction of the governing linear equations instead of Jacobi polynomials. A simple strategy is suggested for the evaluation of the required improper integrals. Near-field results are validated through convergence analysis. Far-field patterns are compared with predictions of the surface equivalence theorem (SET).

Superconvergent postprocessing of the C1-conforming finite element method for fourth-order boundary value problems

We develop a very simple but efficient postprocessing technique for enhancing the accuracy of the C1-conforming finite element method for fourth-order boundary value problems. The key idea of the postprocessing technique is to add certain generalized Jacobi polynomials of degree larger than k to the Galerkin approximation of degree k. We prove that the postprocess improves the order of convergence of the Galerkin approximation under L2- and H2-norms. Numerical experiments are provided to illustrate the theoretical results. We also present computational results for a steady-state problem with to demonstrate the effectiveness of the postprocessing.

Determining an unknown source in a time-fractional diffusion equation based on Jacobi polynomials expansion with a modified Tiknonov regularization

In this paper, we try to recover an unknown source in a time-fractional diffusion equation. In order to overcome the influence of boundary conditions on source conditions, we introduce the Jacobi polynomials to construct the approximation and a modified Tikhonov regularization method is proposed to deal with the illposedness. Error estimates are obtained under a discrepancy principle as the parameter choice rule. Numerical results are also presented to demonstrate the effectiveness of the proposed method.

Diatomic molecular energy spectra and thermodynamic properties of exponential inversely quadratic plus improved deformed exponential-type potential

By employing the Pekeris-type approximation to deal with the centrifugal term, we study the thermodynamic properties and approximate analytical solutions of the Schrödinger equation with the exponential inversely quadratic plus improved deformed exponential-type potential using parametric Nikiforov–Uvarov method and hypergeometric functional analysis method. The complex eigen energy equation was presented in a closed and compact form and extended to study partition function and other thermodynamic properties. The total normalized wave functions were also obtained and expressed in terms of hypergeometric Jacobi polynomial. The mathematical accuracy of analytical calculation can be verified consistently by comparing numerical data and results of different arbitrarily l state. The potential also reduces to improved deformed exponential-type potential as a special case. We calculated energy spectra for some selected diatomic molecules namely: Silicon Flouride (SiF+ (X1∑+), Oxygen molecule O2+ (X2∏g), Nitrogen molecule, N2+ (X1∑g+), Sulphur chloride SCl and Rubidium hydride RbH using standard molecular spectroscopic constants as applied to the two methods. The results show that the energy eigenvalues for these selected diatomic molecules are in excellent agreement using the two methods and also increases with an increase in quantum state.

Finite elements based on Jacobi shape functions for the analysis of beams, plates and shells

AbstractThis paper proposes the use of Jacobi polynomials to approximate higher‐order theories of beam, plate, and shell structures. The Carrera unified formulation is used in this context to express displacement kinematics in a hierarchical form. In this manner, classical to complex higher‐order theories can be implemented with ease. Particular attention is focused on the attenuation and the correction of the shear locking. Therefore, reduced integration as well as mixed interpolation of tensorial components methods are investigated against the new finite elements. Several case studies are taken into account to highlight the effectiveness and robustness of the proposed approach. Also, several benchmarks are provided for future assessments.

CERTAIN UNIFIED INTEGRALS INVOLVING A PRODUCT OF THE FOUR-PARAMETER BESSEL FUNCTION AND JACOBI POLYNOMIAL

The present paper is devoted to derive a generalized Oberhettingertype integral formula. The derived form of an integral involving the product of the four-parameter Bessel function and Jacobi polynomial. The outcomes are expressed in terms of the Kamp´e de F´eriet and Srivastava and Daoust functions. Also, four Corollaries of the both Theorems are derived in terms of the Kamp´e de F´eriet and Srivastava and Daoust functions. Some of the significant particular cases are also determined. Furthermore, we drive an interesting relationship between Kamp´e de F´eriet and Srivastava and Daoust functions

Generalised tanh-shaped hyperbolic potential: Klein–Gordon equation's bound state solution

The development of potential theory heightens the understanding of fundamental interactions in quantum systems. In this paper, the bound state solution of the modified radial Klein–Gordon equation is presented for generalised tanh-shaped hyperbolic potential from the Nikiforov–Uvarov method. The resulting energy eigenvalues and corresponding radial wave functions are expressed in terms of the Jacobi polynomials for arbitrary l states. It is also demonstrated that energy eigenvalues strongly correlate with potential parameters for quantum states. Considering particular cases, the generalised tanh-shaped hyperbolic potential and its derived energy eigenvalues exhibit good agreement with the reported findings. Furthermore, the rovibrational energies are calculated for three representative diatomic molecules, namely H2, HCl and O2. The lowest excitation energies are in perfect agreement with experimental results. Overall, the potential model is displayed to be a viable candidate for concurrently prescribing numerous quantum systems.

Discrete harmonic analysis associated with Jacobi expansions III: The Littlewood–Paley–Stein [formula omitted]-functions and the Laplace type multipliers

The research about harmonic analysis associated with Jacobi expansions carried out in Arenas et al. (2020) and Arenas et al. (2022) is continued in this paper. Given the operator J(α,β)=J(α,β)−I, where J(α,β) is the three-term recurrence relation for the normalized Jacobi polynomials and I is the identity operator, we define the corresponding Littlewood–Paley–Stein gk(α,β)-functions associated with it and we prove an equivalence of norms with weights for them. As a consequence, we deduce a result for Laplace type multipliers.

Exceptional Jacobi polynomials which are deformations of Jacobi polynomials

In [10] we constructed new examples of exceptional Jacobi polynomials. The difference with the so called standard families is that the examples constructed in [10] depend on an arbitrary number of continuous parameters. Besides these continuous parameters, denoted by M={M0,M1,…}, each one of the new families of exceptional Jacobi polynomials is associated to two negative integers a≤b≤−1 and a finite set of positive integers F satisfying that {−b,…,−a−b−1}⊂F. In this paper we prove that when F={u,u+1,…,−a−b−1}, with 1≤u≤−b, the exceptional Jacobi polynomials are deformations of Jacobi polynomials (when the continuous parameters Mi→1). Otherwise, the exceptional Jacobi polynomials are deformations of standard exceptional Jacobi polynomials (again when the continuous parameters Mi→1).

Overlapping of Lévai’s and Milson’s e-Tangent-Polynomial Potentials along Symmetric Curves

The paper examines common elements between Lévai’s and Milson’s potentials obtained by Liouville transformations of two rational canonical Sturm–Liouville equations (RCSLEs) with even density functions which are exactly solvable via Jacobi polynomials in a real or accordingly imaginary argument. We refer to the polynomial numerators of the given rational density function as ‘tangent polynomial’ (TP) and thereby term the aforementioned potentials as ‘e-TP’. Special attention is given to the overlap between the two potentials along symmetric curves which represent two different rational forms of the Ginocchio potential exactly quantized via Gegenbauer and Masjed-Jamei polynomials, respectively. Our analysis reveals that the actual interconnection between Lévai’s parameters for these two rational realizations of the Ginocchio potential is much more complicated than one could expect based on the striking resemblance between two quartic equations derived by Lévai for ‘averaged’ Jacobi indexes.

Parametrization and thermodynamic scaling of pair correlation functions for the fractional quantum Hall effect

The calculation of pair correlations and density profiles of quasiholes are routine steps in the study of proposed fractional quantum Hall states. Nevertheless, the field has not adopted a standard way to present the results of such calculations in an easily reproducible form. We develop a polynomial expansion that allows for easy quantitative comparison between different candidate wavefunctions, as well as reliable scaling of correlation and quasihole profiles to the thermodynamic limit. We start from the well-known expansion introduced by Girvin [PRB, 30 (1984)] (see also [Girvin, MacDonald and Platzman, PRB, 33 (1986)]), which is physically appealing but, as we demonstrate, numerically unstable. We orthogonalize their basis set to obtain a new basis of modified Jacobi polynomials, whose coefficients can be stably calculated. We then apply our expansion to extract pair correlation expansion coefficients and quasihole profiles in the thermodynamic limit for a wide range of fractional quantum Hall wavefunctions. These include the Laughlin series, composite fermion states with both reverse and direct flux attachment, the Moore-Read Pfaffian state, and BS hierarchy states. The expansion procedure works for both abelian and non-abelian quasiholes, even when the density at the core is not zero. We find that the expansion coefficients for all quantum Hall states considered can be fit remarkably well using a cosine oscillation with exponentially decaying amplitude. The frequency and the decay length are related in an intuitive, but not elementary way to the filling fraction. Different states at the same filling fraction can have distinct values for these parameters. Finally, we also use our scaled correlation functions to calculate estimates for the magneto-roton gaps of the various states.

Gyroscopic polynomials

Gyroscopic alignment gives rise to highly spatially anisotropic columnar structures that in combination with complex domain boundaries pose challenges for efficient numerical discretizations and computations. We define gyroscopic polynomials to be three-dimensional polynomials expressed in a coordinate system that conforms to rotational alignment. We remap the original domain with radius-dependent boundaries onto a right cylindrical or annular domain to create the computational domain in this coordinate system. We find the volume element expressed in gyroscopic coordinates leads naturally to a hierarchy of orthonormal bases. We build the bases out of Jacobi polynomials in the vertical and generalized Jacobi polynomials in the radial. Because these coordinates explicitly conform to flow structures found in rapidly rotating systems the bases represent fields with a relatively small number of modes. We develop the operator structure for one-dimensional semi-classical orthogonal polynomials as a building block for differential operators in the full three-dimensional cylindrical and annular domains. The differentiation operators of generalized Jacobi polynomials generate a sparse linear system for discretization of differential operators acting on the gyroscopic bases. This enables efficient simulation of systems with strong gyroscopic alignment.

The bilateral birth–death chain generated by the associated Jacobi polynomials

AbstractWe give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three‐term recurrence relation for the classical Jacobi polynomials by shifting the integer index n by a real number t. Under certain restrictions, this will give rise to a doubly infinite tridiagonal stochastic matrix, which can be interpreted as the one‐step transition probability matrix of a discrete‐time bilateral birth–death chain with state space on . We also study the unique UL and LU stochastic factorizations of the transition probability matrix, as well as the discrete Darboux transformations and corresponding spectral matrices. Finally, we use all these results to provide an urn model on the integers for the associated Jacobi polynomials.

Finite elements based on Jacobi shape functions for the free vibration analysis of beams, plates, and shells

This article proposes the use of Jacobi polynomials as shape functions for the free vibration analysis of beam, plate, and shell structures. Jacobi polynomials, indicated as P p ( γ , θ ) , belong to the family of classical orthogonal polynomials, and depend on two scalar parameters γ and θ, with p being the polynomial order. The Jacobi-like shape functions are built in the context of the Carrera unified formulation, which permits the expression of displacement kinematics in a hierarchical form. In this manner, it is possible to adopt several classical to complex higher-order theories with ease. Particular attention is focused on the attenuation and the correction of the shear locking. The results have been compared with analytical results from the literature. For the plate benchmark, analytical results are introduced as the reference results in this article for the first time using the closed form of CUF. Beams, plates, and shells with different thicknesses have been considered. It is demonstrated that the parameters γ and θ are not influential for the calculations.

Classical Orthogonal Polynomials Revisited

This manuscript contains a small portion of the algebraic theory of orthogonal polynomials developed by Maroni and their applicability to the study and characterization of the classical families, namely Hermite, Laguerre, Jacobi, and Bessel polynomials. It is presented a cyclical proof of some of the most relevant characterizations, particularly those due to Al-Salam and Chihara, Bochner, Hahn, Maroni, and McCarthy. Two apparently new characterizations are also added. Moreover, it is proved through an equivalence relation that, up to constant factors and affine changes of variables, the four families of polynomials named above are the only families of classical orthogonal polynomials.