Generalized Gegenbauer Polynomials Research Articles

A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS

In this paper, we consider the following \(\mathcal{L}\)-difference equation$$\Phi(x) \mathcal{L}P_{n+1}(x)=(\xi_nx+\vartheta_n)P_{n+1}(x)+\lambda_nP_{n}(x),\quad n\geq0,$$where \(\Phi\) is a monic polynomial (even), \(\deg\Phi\leq2\), \(\xi_n,\,\vartheta_n,\,\lambda_n,\,n\geq0\), are complex numbers and \(\mathcal{L}\) is either the Dunkl operator \(T_\mu\) or the the \(q\)-Dunkl operator \(T_{(\theta,q)}\). We show that if \(\mathcal{L}=T_\mu\), then the only symmetric orthogonal polynomials satisfying the previous equation are, up a dilation, the generalized Hermite polynomials and the generalized Gegenbauer polynomials and if \(\mathcal{L}=T_{(\theta,q)}\), then the \(q^2\)-analogue of generalized Hermite and the \(q^2\)-analogue of generalized Gegenbauer polynomials are, up a dilation, the only orthogonal polynomials sequences satisfying the \(\mathcal{L}\)-difference equation.

Eighth-Kind Chebyshev Polynomials Collocation Algorithm for the Nonlinear Time-Fractional Generalized Kawahara Equation

In this study, we present an innovative approach involving a spectral collocation algorithm to effectively obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation (NTFGKE). We introduce a new set of orthogonal polynomials (OPs) referred to as “Eighth-kind Chebyshev polynomials (CPs)”. These polynomials are special kinds of generalized Gegenbauer polynomials. To achieve the proposed numerical approximations, we first derive some new theoretical results for eighth-kind CPs, and after that, we employ the spectral collocation technique and incorporate the shifted eighth-kind CPs as fundamental functions. This method facilitates the transformation of the equation and its inherent conditions into a set of nonlinear algebraic equations. By harnessing Newton’s method, we obtain the necessary semi-analytical solutions. Rigorous analysis is dedicated to evaluating convergence and errors. The effectiveness and reliability of our approach are validated through a series of numerical experiments accompanied by comparative assessments. By undertaking these steps, we seek to communicate our findings comprehensively while ensuring the method’s applicability and precision are demonstrated.

A New Generalization of Gegenbauer Polynomials

In this work, the author introduces new generalization of Gegenbauer polynomials of one and two variables by considering new extended gamma function defined by MacDonald function. Certain properties of this new generalized Gegenbauer polynomials like integral formulas, Mellin transform, recurrence relationsand generating function are presented and investigated.

Characterization of q-Dunkl-classical symmetric orthogonal q-polynomials

In this paper, we show that, up to a dilatation, the $$q^2$$ -analogue of generalized Hermite and $$q^2$$ -analogue of generalized Gegenbauer polynomials are the only q-Dunkl-classical symmetric orthogonal polynomials.

Some results on Dunkl-coherent Pairs

ABSTRACT We introduce in this paper the concept of Dunkl-coherent pair of forms (linear functionals) in the symmetric case. We prove that if is a Dunkl-symmetrically coherent pair of symmetric forms, then at least one of them has to be a symmetric Dunkl-classical form, i.e. Generalized Hermite or Generalized Gegenbauer polynomials.

On the smoothness of multi center coplanar black hole and membrane horizons

We study the differentiability of the metric and other fields at any of the horizons of multi center Reissner–Nordstrom black hole solutions in $$d \ge 5$$ and of multi center M2 brane solutions. The centers are distributed in a plane in transverse space, hence termed coplanar. We construct the Gaussian null co-ordinate system for the neighborhood of a horizon by solving the geodesic equations in expansions of (appropriate powers of) the affine parameter. Organizing the harmonic functions that appear in the solution in terms of what can be called generalized Gegenbauer polynomials is key to obtaining the solution to the geodesic equations in a compact and manageable form. We then compute the metric and other fields in the Gaussian null co-ordinate system and find that the differentiability of the coplanar solution is identical to the differentiability of the collinear solution (centers distributed on a line in transverse space). We end the paper with a conjecture on the degree of smoothness of the most general multi center solution, the one with centers distributed arbitrarily.

Some identities involving generalized Gegenbauer polynomials

In this paper, we investigate some interesting identities on the Bernoulli, Euler, Hermite and generalized Gegenbauer polynomials arising from the orthogonality of generalized Gegenbauer polynomials in the generalized inner product \t\t\t〈p1(x),p2(x)〉=∫−αqpαqp(αq−p2x2)λ−12p1(x)p2(x)dx.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\bigl\\langle {{p_{1}}(x),{p_{2}}(x)} \\bigr\\rangle = \\int_{ - \\frac{{\\sqrt{\\alpha q}}}{p}}^{\\frac{{\\sqrt{ \\alpha q} }}{p}} {{\\bigl(\\alpha q - p^{2}{x^{2}}\\bigr)}^{\\lambda - \\frac{1}{2}}} {p_{1}}(x){p_{2}}(x)\\,dx. $$\\end{document}

On parameter derivatives of a family of polynomials in two variables

The purpose of the present paper is to give the parameter derivative representations of the form∂Pn,k(λ;x,y)∂λ=∑m=0n-1∑j=0mdn,j,mPm,j(λ;x,y)+∑j=0ken,j,kPn,j(λ;x,y)for a family of orthogonal polynomials of variables x and y, with λ being a parameter and 0⩽k⩽n;n,k=0,1,2,…. First, we shall present the representations of the parameter derivatives of the generalized Gegenbauer polynomials Cn(λ,μ)(x) with the help of the parameter derivatives of the classical Jacobi polynomials Pn(α,β)(x), i.e. ∂∂αPn(α,β)(x) and ∂∂βPn(α,β)(x). Then, by using these derivatives, we investigate the parameter derivatives for two-variable analogues of the generalized Gegenbauer polynomials. Furthermore, we discuss orthogonality properties of the parametric derivatives of these polynomials.

Characterization of the generalized Gegenbauer polynomials

We characterize the generalized Gegenbauer polynomials, then we provide a closed form of the the generalized Gegenbauer polynomials using Bernstein basis. We conclude the paper with some results concerning integrals of the generalized Gegenbauer and Bernstein basis. Mathematics Subject Classification: 33C50, 33C45, 05A10, 33B15

A characterization of symmetric Tμ-classical monic orthogonal polynomials by a structure relation

The aim of this work is to study the symmetric orthogonal polynomials’ sequences which are Dunkl-classical. We recover again that the generalized Hermite polynomials and the generalized Gegenbauer polynomials are the only symmetric Dunkl-classical through a different manner that in Ben Cheikh and Gaied [Characterization of the Dunkl-classical symmetric orthogonal polynomials. Appl Math Comput. 2007;187:105–114] by solving a differential-difference equation in the dual space of polynomials. Moreover, we deduce the structure relation that such a symmetric monic orthogonal polynomials satisfies.

Some new results about a set of semi-classical polynomials of class s=1

We establish some properties concerning the semi-classical sequence of monic orthogonal polynomials of class s=1, verifying the three-recurrence relation with β n =(−1) n β0, n≥0, where β0∈ℂ−{0}. We write this sequence of polynomials as simple summations. An integral representation of the corresponding linear functional is given. A connection with the Jacobi and the generalized Gegenbauer polynomials is presented.

Investigation of continuous-time quantum walk on root lattice [formula omitted] and honeycomb lattice

The continuous-time quantum walk (CTQW) on root lattice A n (known as hexagonal lattice for n = 2 ) and honeycomb one is investigated by using spectral distribution method. To this aim, some association schemes are constructed from abelian group Z m ⊗ n and two copies of finite hexagonal lattices, such that their underlying graphs tend to root lattice A n and honeycomb one, as the size of the underlying graphs grows to infinity. The CTQW on these underlying graphs is investigated by using the spectral distribution method and stratification of the graphs based on Terwilliger algebra, where we get the required results for root lattice A n and honeycomb one, from large enough underlying graphs. Moreover, by using the stationary phase method, the long time behavior of CTQW on infinite hexagonal lattice is approximated with finite ones for finite distances from the origin while for large distances, the scaling behavior of the probability distribution is deduced. Also, it is shown that, the probability amplitudes at finite times possess the point group symmetry in the sense that two lattice points have the same probability amplitudes if and only if they belong to the same orbit (stratum of the graph) of the point group of the lattice. Apart from physical results, it is shown that the Bose–Mesner algebras of our constructed association schemes (called n-variable P-polynomial) can be generated by n commuting generators, where raising, flat and lowering operators (as elements of Terwilliger algebra) are associated with each generator. A system of n-variable orthogonal polynomials which are special cases of generalized Gegenbauer polynomials is constructed, where the probability amplitudes are given by integrals over these polynomials or their linear combinations. Finally the supersymmetric structure of finite honeycomb lattices is revealed.

Characterization of the Dunkl-classical symmetric orthogonal polynomials

In this paper, we introduce the notion of Dunkl-classical orthogonal polynomials. Then, we show that generalized Hermite and generalized Gegenbauer polynomials are the only Dunkl-classical symmetric orthogonal polynomials by solving a suitable differential-difference equation.

On Generating Relations Involving Generalized Gegenbauer Polynomials

Abstract In this paper, generating relations involving generalized Gegenbauer polynomials are obtained by constructing a three-dimensional Lie algebra isomorphic to special linear algebra sl(2). Further, a number of new interesting relations involving various generalized polynomials are obtained as applications of these generating relations.

An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials

The generalized Gegenbauer polynomials are orthogonal polynomials with respect to the weight function | x| 2 μ (1− x 2) λ−1/2 . An integral formula for these polynomials is proved, which serves as a transformation between h-harmonic polynomials associated with Z 2 invariant weight functions on the plane. The formula also gives a new integral transform for the Jacobi polynomials, which contains several well-known formulae as special cases. The new formulae can be used to prove the positivity of certain sums of the generalized Gegenbauer and Jacobi polynomials.

DERIVATIVES OF GENERALIZED GEGENBAUER POLYNOMIALS

We prove some new formulas for the derivatives of the generalized Gegenbauer polynomials associated with the Lie algebra A2.

The affine group and generalized Gegenbauer polynomials

Operators of the form f( xD) — g( L), where L is a shift (lowering) operator, arise naturally in the study of stochastic processes, such as Brownian motion, on the affine group. We find the polynomial eigenfunctions and the action of the affine group as well as the matrix elements of an exponential function corresponding to L. Then it is determined when the eigenfunctions considered form a family of orthogonal polynomials.

Bessel Functions of Matrix Argument

Our principal results fall into three main classes. First, a large number of formulae from the classical theory of special functions are given appropriate generalizations. Some of these turn out to have applications to lattice-point problems and to the theory of non-central Wishart distributions in statistics. Secondly, the L2-theory of the Hankel transform is established with the generalized Bessel functions furnishing the kernel, i.e. the transformation g(A) = JM>oA y(AM)f(M) (det M)7 dM is a self-reciprocal unitary correspondence of the Hilbert space of functions for which f A>O I f(A) 12 (det A)7 dA < c* onto itself. Here Py is a real number greater than -1 and A and M are positive definite matrices. In this connection there are two results we wish to emphasize. (1) A complete set of eigenfunctions for the Hankel transform is given in the form' etr(- A)L(') (2A), t running over a certain index class, with the L() (A) as polynomials in the entries of the matrix A. These polynomials enjoy generalized versions of nearly all the properties of the Laguerre polynomials to which they reduce in the scalar case. (2) The ordinary multi-dimensional Fourier transform of a function of mk variables satisfying a certain generalized radiality condition reduces to a Hankel transform. More precisely, arrange the mk variables in a k X m matrix T; then if the function depends only on R = T'T, T' being the transposed matrix, the Fourier transform may be computed in terms of the Hankel transform of order -y = 2 (k - m - 1) defined for functions of positive semidefinite m X m matrices R. The third class of results concerns the properties of harmonic polynomials in several variables having a certain matrix homogeneity. We call a polynomial, P(T), in the entries of the k X m matrix T, an H-polynomial if (1) P(T) is a harmonic function of mk variables and (2) P(TZ) = (det Z)VP(T) for some integer v and all m X m symmetric matrices Z. These H-polynomials behave like Stieffel-manifold (in contrast to spherical) harmonics. They are related in a natural way to generalized Gegenbauer polynomials which are in turn defined as hypergeometric functions.