Other ways of defining orthogonal polynomials include differentiation formulas and generating functions. Pdf simple approach to gegenbauer polynomials researchgate. In section 2, we introduce basic notation and denitions. The connection relation for jacobi polynomials with two free. Pdf approximation by gegenbauer polynomials in the study. Algebraic generating functions for gegenbauer polynomials. In this work, we develop fast algorithms for computations involving nite expansions in gegenbauer polynomials. Tom koornwinder dual addition formula for gegenbauer polynomials. In mathematics, gegenbauer polynomials or ultraspherical polynomials c. Computing with expansions in gegenbauer polynomials jens keiner abstract. There has to be a good reason why the gegenbauer polynomials were also named ultraspherical polynomials. The properties of the gegenbauer polynomials and the representation will play the key role. But that is as far as i know how to take that reasoning. The gegenbauer polynomials are orthogonal on the interval with weight function, corresponding to integration over a unit hypersphere.
We show that the only orthogonal polynomials with a generating function of the form f xz z2 are the ultraspherical, hermite, and chebyshev polynomials of the. Clifford gegenbauer polynomials related to the dunkl dirac operator. Orthogonal polynomials are sometimes defined using the differential equations they satisfy as functions of or the recurrence relations they satisfy with respect to the order. The principal aim of this paper is to develop, implement and analyse an e. A new family of scaling and wavelet functions is introduced. The link of ordinary 2nd order differential equations to multiresolution filters is employed to construct these new functions. The gegenbauer polynomials, and thus also the legendre, zernike and chebyshev polynomials, are special cases of the jacobi polynomials. The appropriate choice of the free parame ter enables us to generate filter functions obtained with chebyshev polynomials of the first and second kind and. It is shown that several of brafmans generating functions for the gegenbauer polynomials are algebraic functions of their arguments, if the gegenbauer. Find the value of the fifthdegree gegenbauer polynomial for the parameter a at these points.
A family of new generating functions for the gegenbauer polynomials is presented. In this paper, we derive some interesting identities involving gegenbauer polynomials arising from the orthogonality of gegenbauer polynomials for the inner product space p n with respect to the weighted inner product. The formula also gives a new integral transform for the jacobi polynomials, which. This result establishes a connection between uniform estimates for 1. This paper is a modest attempt to demonstrate the applicability of generalized. Mathematics differential equations research mathematical research polynomials topological spaces. Dilcher, zerofree regions for bernoulli polynomials, c. Reconstruction of sparse legendre and gegenbauer expansions 1021. Application to the analysis of a waveguide septum polarizer. Interested readers may refer to 1,2,9 for full accounts of orthogonal polynomials and also to 14,15 for papers discussing relevant orthogonal polynomials. The connection is new and is a direct consequence of wellknown properties of gegenbauer polynomials. Milovanovi c university of ni s, faculty of technology leskovac, 2014. G e g e n b a u e r p o l y n o m i a l s a r e r e l a t e d t o t nx, t h e chebyshev polynomials of the first kind, and to u nx, the chebyshev polynomials of the second kind, by t h e r e l a t i o n s t n x l i m l n j j n i n l, 1.
Connection problem for sums of finite products of legendre. We describe a method to convert a linear combination of gegenbauer polynomials up to degree ninto a representation in a di erent family of gegenbauer polynomials. Developed with mathematica download a free trial version. This work is based upon the elementary manipulation of series and is motivated by the recent appearance of these polynomials in certain aspects of applied mathematics. Cohl 7 september 2010 a thesis submitted in fulfilment of the requirements for the degree of doctor of philosophy in mathematics at the university of auckland, 2010. Media in category gegenbauer polynomials the following 7 files are in this category, out of 7 total. We also connect our weight matrix w p,n with the weight considered in 15, where the authors give examples of matrix valued gegenbauer polynomials, extending for an arbitrary parameter. Fourier and gegenbauer expansions for fundamental solutions of the laplacian and powers in rdand hd howard s. Reconstruction of sparse legendre and gegenbauer expansions. Depending on its arguments, gegenbauerc returns floatingpoint or exact symbolic results. The gegenbauer polynomials and typically real functions. Report by bulletin of the belgian mathematical society simon stevin. In this section, we show that the conversion between different expansions in terms of gegenbauer polynomials corresponds to the eigenvector matrix of an explicitly known umatrix. The gegenbauer polynomials include a number of polynomials we have seen previously as special cases.
Gegenbauer polynomials cl n x, and jacobi polynomials p a,b n x. Gegenbauer polynomials in mathematics, gegenbauer polynomials or ultraspherical polynomials c n. Apart from constant factors they are certain jacobi polynomials. An integral formula for generalized gegenbauer polynomials. Fourier, gegenbauer and jacobi expansions for a power. Pdf algebraic generating functions for gegenbauer polynomials. Also known as ultraspherical polynomials explanation of gegenbauer polynomial. Operator methods and su1,1 symmetry in the theory of. Approximation by gegenbauer polynomials in the study of a rectangular ridged waveguide.
Filter function synthesis by gegenbauer generating function 61. As a matter of fact, this was the primary reason for introducing generalized polynomials in the rst place cf. For certain special arguments, gegenbauerc automatically evaluates to exact values. A generalization of the generating function for gegenbauer polynomials is introduced whose coefficients are given in terms of associated legendre functions of the second kind. Pdf some identities involving gegenbauer polynomials. In section 3, we briey survey chandrasekarans and gus divideand conquer. Gegenbauer polynomials and semiseparable matrices 3 gantmakher and krein, statistical analysis with covariance matrices that turn out to be specially structured graybill 16, and the discretisation of certain integral. Gegenbauer polynomials for numeric and symbolic arguments. In short form one finds a lot of formulae for orthogonal polynomials also in a book of bell which we cite mainly for a formula to hermite poly22 nomials see footnote in section 5.
We discuss how our expansion represents a generalization of several previously derived formulae such as heines formula and heines reciprocal squareroot identity. Letter section generating functions for gegenbauer polynomials. Gegenbauer polynomials, polynomial transforms, fast algorithms 1. An integral formula for these polynomials is proved, which serves as a transformation between hharmonic polynomials associated with z 2 invariant weight functions on the plane. The generating function for the chebyshev case is nonstandard, although it is easily derived.
Ultraspherical polynomials encyclopedia of mathematics. Cliffordgegenbauer polynomials related to the dunkl dirac. They generalize legendre polynomials and chebyshev polynomials, and are special cases of jacobi polynomials. In mathematics, jacobi polynomials occasionally called hypergeometric polynomials p. Gegenbauercn,lambda,z 205 formulas primary definition 1 formula specific values 26 formulas general characteristics 10 formulas series representations 56 formulas integral representations 2 formulas generating functions 1 formula. The main result which this paper relies upon is theorem2. Varying the free parameter v, we can control the value of the maximum amplification in the passband or the slope in. In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of obrechko and certain explicit connection coe cients for these polynomials. On a generalization of the generating function for. A family of polynomials solving a special case of the gauss hypergeometric equation.
The ultraspherical polynomials are the coefficients of the power series expansion of the generating function the ultraspherical polynomial satisfies the differential equation more commonly used are the formulas for references see orthogonal polynomials. Gegenbauer polynomials, we show that the corresponding linear mapping is the eigenvector matrix of an explicitly known diagonal plus upper triangular semiseparable matrix. The generalized gegenbauer polynomials are orthogonal polynomials with respect to the weight function x 2. Dimitrov, eduardo godoy, and andr e ronveaux abstract. Pdf gegenbauer polynomials are obtained through well known linear algebra methods based on sturmliouville theory. Preface in this book we collect several recent results on special classes of polynomi. The tnx are ops on 1,1 with respect to the weight function 1 x2 12. Filter function synthesis by gegenbauer generating function. In the case of some linear functionals the problem has an easy solution thanks to the kreinmillman theorem and the simple form of the extreme points of t r. Pdf in this paper we derive some interesting identities arising from the orhtogonality of gegenbauer polynomials.
Generalizations of the classical gegenbauer polynomials to the clifford analysis framework are called clifford gegenbauer polynomials and were introduced as well on the closed unit ball b1 see 4, as on the euclidean space r. We derive a generalization of the generating function for gegenbauer polynomials through extension a two element sequence of generating functions for jacobi polynomials. Because these numbers are not symbolic objects, gegenbauerc returns floatingpoint results. These classes are named as polynomials of legendre, gegen. In the constructive theory of spherical functions the gegenbauer polynomials play an important role. Gegenbauerc can be evaluated to arbitrary numerical precision. Orthogonal polynomials 75 where the yij are analytic functions on c \ r, and solve for such matrices the following matrixvalued riemannhilbert problem.
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