New developments for the Jacobi polynomials

This review is dedicated to Jacobi polynomials which are a generalization of some well-known classical polynomials such as Gegenbauer polynomials, Legendre polynomials, first and second kinds Chebyshev polynomials and Zernike polynomials. To save effort and time, it is advantageous and sufficient to investigate the properties of Jacobi polynomials rather than considering their special cases separately. For demonstration purposes, we show how to reduce most of the obtained properties of Jacobi polynomials to the corresponding properties of Legendre polynomials such as the generating function, Rodrigues formula, special values and the orthogonality property. Most of the properties of Jacobi polynomials are obtained through their hypergeometric representations such as differential recurrence relations, generating functions, some special values (exact and asymptotic) and some integral expansions of positive integrand. The standard orthogonality property of Jacobi polynomials for the values of the parameters is discussed and some applications of such important property are pointed out. The orthogonality property of Jacobi polynomials considerably affects the positions of their zeros. These zeros are essential in any type of numerical quadrature such as Legendre-Gaussian quadrature, first kind Chebyshev-Gaussian quadrature and second kind Chebyshev-Gaussian quadrature. Moreover the recurrence relations of Jacobi polynomials play an important role in computing the zeros of Jacobi polynomials which are needed in the Gaussian-Jacobi quadrature. A narrative review is provided on some attempts were done in extending the orthogonality property of Jacobi polynomials to non-standard values of the indexes by amending some of the orthogonality conditions such as Sobolev and non-hermitian orthogonality. Integral expansions of Jacobi polynomials with a prominent feature (positive integrand) were presented in terms of other Jacobi polynomials. Such integral expansions should allow a variety of applications for Jacobi polynomials. This review is dedicated to Jacobi polynomials which are a generalization of some well-known classical polynomials such as Gegenbauer polynomials, Legendre polynomials, first and second kinds Chebyshev polynomials and Zernike polynomials. To save effort and time, it is advantageous and sufficient to investigate the properties of Jacobi polynomials rather than considering their special cases separately. For demonstration purposes, we show how to reduce most of the obtained properties of Jacobi polynomials to the corresponding properties of Legendre polynomials such as the generating function, Rodrigues formula, special values and the orthogonality property. Most of the properties of Jacobi polynomials are obtained through their hypergeometric representations such as differential recurrence relations, generating functions, some special values (exact and asymptotic) and some integral expansions of positive integrand. The standard orthogonality property of Jacobi polynomials for the values of the parameters is discussed and some applications of such important property are pointed out. The orthogonality property of Jacobi polynomials considerably affects the positions of their zeros. These zeros are essential in any type of numerical quadrature such as Legendre-Gaussian quadrature, first kind Chebyshev-Gaussian quadrature and second kind Chebyshev-Gaussian quadrature. Moreover the recurrence relations of Jacobi polynomials play an important role in computing the zeros of Jacobi polynomials which are needed in the Gaussian-Jacobi quadrature. A narrative review is provided on some attempts were done in extending the orthogonality property of Jacobi polynomials to non-standard values of the indexes by amending some of the orthogonality conditions such as Sobolev and non-hermitian orthogonality. Integral expansions of Jacobi polynomials with a prominent feature (positive integrand) were presented in terms of other Jacobi polynomials. Such integral expansions should allow a variety of applications for Jacobi polynomials.

Next article Recurrence Relations for the Coefficients in Jacobi Series Solutions of Linear Differential EquationsStanisław LewanowiczStanisław Lewanowiczhttps://doi.org/10.1137/0517074PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA method is presented for obtaining recurrence relations for the coefficients in Jacobi series solutions of linear ordinary differential equations with polynomial coefficients.[1] C. W. Clenshaw, The numerical solution of linear differential equations in Chebyshev series, Proc. Cambridge Philos. Soc., 53 (1957), 134–149 18,516a 0077.32503 CrossrefGoogle Scholar[2] David Elliott, The expansion of functions in ultraspherical polynomials, J. Austral. Math. Soc., 1 (1959/1960), 428–438 23:A1997 0099.28603 CrossrefGoogle Scholar[3] A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York, 1953 Google Scholar[4] L. Fox, Chebyshev methods for ordinary differential equations, Comput. 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Sci., Pretoria, 1979 Google Scholar[17] Jet Wimp, Computation with recurrence relations, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984xii+310 85f:65001 Google ScholarKeywordsJacobi seriesJacobi coefficientsrecurrence relationsdifference operatorslinear differential equation Next article FiguresRelatedReferencesCited byDetails Descriptions of fractional coefficients of Jacobi polynomial expansions18 April 2022 | The Journal of Analysis, Vol. 30, No. 4 Cross Ref On Jacobi polynomials and fractional spectral functions on compact symmetric spaces4 January 2021 | The Journal of Analysis, Vol. 29, No. 3 Cross Ref Spectral Solutions for Differential and Integral Equations with Varying Coefficients Using Classical Orthogonal Polynomials17 July 2018 | Bulletin of the Iranian Mathematical Society, Vol. 45, No. 2 Cross Ref On the coefficients of differentiated expansions and derivatives of chebyshev polynomials of the third and fourth kindsActa Mathematica Scientia, Vol. 35, No. 2 Cross Ref On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials6 January 2004 | Journal of Physics A: Mathematical and General, Vol. 37, No. 3 Cross Ref On the coefficients of differentiated expansions and derivatives of Jacobi polynomials8 April 2002 | Journal of Physics A: Mathematical and General, Vol. 35, No. 15 Cross Ref The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable functionJournal of Computational and Applied Mathematics, Vol. 89, No. 1 Cross Ref On the legendre coefficients of the moments of the general order derivative of an infinitely differentiable functionInternational Journal of Computer Mathematics, Vol. 56, No. 1-2 Cross Ref Evaluation of Bessel function integrals with algebraic singularitiesJournal of Computational and Applied Mathematics, Vol. 37, No. 1-3 Cross Ref Properties of the polynomials associated with the Jacobi polynomials1 January 1986 | Mathematics of Computation, Vol. 47, No. 176 Cross Ref Volume 17, Issue 5| 1986SIAM Journal on Mathematical Analysis History Submitted:15 April 1985Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsKeywordsJacobi seriesJacobi coefficientsrecurrence relationsdifference operatorslinear differential equationMSC codes42C1039A7065L0565L10PDF Download Article & Publication DataArticle DOI:10.1137/0517074Article page range:pp. 1037-1052ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

We introduce the so-called Clifford-Gegenbauer polynomials in the framework of Dunkl operators, as well on the unit ball B(1), as on the Euclidean space IR(m). In both cases we obtain several properties of these polynomials, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the Jacobi polynomials on the real line. As in the classical Clifford case, the orthogonality of the polynomials on IR(m) must be treated in a completely different way than the orthogonality of their counterparts on B(1). In case of IR(m), it must be expressed in terms of a bilinear form instead of an integral. Furthermore, in this paper the theory of Dunkl monogenics is further developed.

In this paper, we introduce and study the new family of analytic functions via Rodrigues formula. Some main properties, the generating function, various recurrence relations and differential properties of these functions are obtained. Furthermore, the differential equations are given for the subclasses of this family of analytic functions.

The present review discusses the solution of the Heun, confluent, biconfluent, double confluent, and triconfluent equations in terms of polynomial expansions, and applies the results to generate exactly solvable Schrödinger potentials. Although there are more general approaches to solve these differential equations in terms of the expansions of certain special functions, the importance of polynomial solutions is unquestionable, as most of the known potentials are solvable in terms of the hypergeometric and confluent hypergeometric functions; i.e., Natanzon-class potentials possess bound-state solutions in terms of classical orthogonal polynomials, to which the (confluent) hypergeometric functions can be reduced. Since some of the Heun-type equations contain the hypergeometric and/or confluent hypergeometric differential equations as special limits, the potentials generated from them may also contain Natanzon-class potentials as special cases. A power series expansion is assumed around one of the singular points of each differential equation, and recurrence relations are obtained for the expansion coefficients. With the exception of the triconfluent Heun equations, these are three-term recurrence relations, the termination of which is achieved by prescribing certain conditions. In the case of the biconfluent and double confluent Heun equations, the expansion coefficients can be obtained in the standard way, i.e., after finding the roots of an (N + 1)th-order polynomial in one of the parameters, which, in turn, follows from requiring the vanishing of an (N + 1) × (N + 1) determinant. However, in the case of the Heun and confluent Heun equations, the recurrence relation can be solved directly, and the solutions are obtained in terms of rationally extended X1-type Jacobi and Laguerre polynomials, respectively. Examples for solvable potentials are presented for the Heun, confluent, biconfluent, and double confluent Heun equations, and alternative methods for obtaining the same potentials are also discussed. These are the schemes based on the rational extension of Bochner-type differential equations (for the Heun and confluent Heun equation) and solutions based on quasi-exact solvability (QES) and on continued fractions (for the biconfluent and double confluent equation). Possible further lines of investigations are also outlined concerning physical problems that require the solution of second-order differential equations, i.e., the Schrödinger equation with position-dependent mass and relativistic wave equations.

Abstract. We consider polynomials which are orthogonal with respect to weight functions, which are defined in terms of the modified Bessel function Iν and which are related to the noncentral χ2 -distribution. It turns out that it is the most convenient to use two weight functions with indices ν and ν+1 and to study orthogonality with respect to these two weights simultaneously. We show that the corresponding multiple orthogonal polynomials of type I and type II exist and give several properties of these polynomials (differential properties, Rodrigues formula, explicit formulas, recurrence relation, differential equation, and generating functions).

A two-variable generalization of the Big -1 Jacobi polynomials is introduced and characterized. These bivariate polynomials are constructed as a coupled product of two univariate Big -1 Jacobi polynomials. Their orthogonality measure is obtained. Their bispectral properties (eigenvalue equations and recurrence relations) are determined through a limiting process from the two-variable Big q-Jacobi polynomials of Lewanowicz and Woźny. An alternative derivation of the weight function using Pearson-type equations is presented.

Connections between two-variable Bernstein and Jacobi polynomials on the triangle

General Background: Wavelet approximations are fundamental in numerical analysis and signal processing, with classical orthogonal polynomials like Jacobi and Chebyshev serving as key tools due to their strong approximation properties. Specific Background: The use of Chebyshev wavelets has been extended through generalized polynomial frameworks, such as Koornwinder’s generalization of Jacobi polynomials, offering more flexibility for function approximation on finite intervals. Knowledge Gap: Despite existing wavelet frameworks, the integration of generalized Jacobi and Chebyshev structures into a unified wavelet approximation scheme remains underexplored. Aims: This study introduces the Generalized Jacobi Chebyshev Wavelet (GJCW) approximation, establishing its theoretical foundations and demonstrating convergence and approximation capabilities. Results: It is shown that for a uniformly bounded function expanded in the GJCW basis, the partial sums yield both convergent and best uniform polynomial approximations. Novelty: The formulation of a new wavelet approximation based on a hybrid of generalized Jacobi and Chebyshev polynomials constitutes a novel contribution, supported by rigorous recurrence relations and multiresolution analysis. Implications: This work enhances the theoretical landscape of wavelet-based function approximation, with potential applications in computational mathematics, signal analysis, and numerical solutions of differential equations. Highlight : Wavelet Construction: The paper defines and constructs generalized Jacobi Chebyshev wavelets using orthogonal polynomials. Approximation Theory: It proves that if the wavelet series converges, then a uniform best polynomial approximation exists. Multiresolution Framework: The approach is grounded in Mallat’s multiresolution analysis, enabling efficient function approximation. Keywords : Jacobi Polynomials, Chebyshev Wavelets, Multiresolution Analysis, Polynomial Approximation, Orthonormal Basis

The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials is derived based on that of the Jacobi polynomial.

Past few years have witnessed a considerable level of research activity in the field of exceptional orthogonal polynomials (EOPs), which are new complete orthogonal polynomial systems, and these are first observed as a result of the development of a direct approach to exact or quasi-exact solvability for spectral problems in quantum mechanics that would go beyond the classical Lie algebraic formulations. We have discovered new EOP families associated to such kind of above systems in the framework of supersymmetric quantum mechanics. We have studied thoroughly some fundamental properties of those EOP families. We also have been able to prove completeness of few such EOP categories in weighted Hilbert space, associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken supersymmetry. Some important key properties of such polynomials, e.g, recurrence relation, Rodrigues formula, ladder operators, differential equations, etc., have been obtained.KeywordsOrthogonal PolynomialsLadder OperatorsBroken SupersymmetryWeighted Hilbert SpaceSupersymmetric Quantum MechanicsThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Multiple Hermite polynomials are an extension of the classical Hermite polynomials for which orthogonality conditions are imposed with respect to r>1 normal (Gaussian) weights w_j(x)=e^{-x^2+c_jx} with different means c_j/2, (1 ≤ j ≤ r). These polynomials have a number of properties, such as a Rodrigues formula, recurrence relations (connecting polynomials with nearest neighbor multi-indices), a differential equation, etc. The asymptotic distribution of the (scaled) zeros is investigated and an interesting new feature happens: depending on the distance between the c_j (1 ≤ j ≤ r), the zeros may accumulate on s disjoint intervals, where 1 ≤ s ≤ r. We will use the zeros of these multiple Hermite polynomials to approximate integrals of the form ∫ f(x) \\exp(-x^2 + c_jx) dx simultaneously for 1 ≤ j ≤ r for the case r=3 and the situation when the zeros accumulate on three disjoint intervals. We also give some properties of the corresponding quadrature weights.

Results on the associated Jacobi and Gegenbauer polynomials

Some discrete multiple orthogonal polynomials

Exceptional Laguerre and Jacobi polynomials p n ( x ) p_n(x) are bispectral, in the sense that as functions of the continuous variable x x , they are eigenfunctions of a second order differential operator and as functions of the discrete variable n n , they are eigenfunctions of a higher order difference operator (the one defined by any of the recurrence relations they satisfy). In this paper, under mild conditions on the sets of parameters, we characterize the algebra of difference operators associated to the higher order recurrence relations satisfied by the exceptional Laguerre and Jacobi polynomials.