Jacobi Polynomials

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Sayed Mohammad Abbas - One of the best experts on this subject based on the ideXlab platform.

Alexei Zhedanov - One of the best experts on this subject based on the ideXlab platform.

  • Two-variable -1 Jacobi Polynomials
    Integral Transforms and Special Functions, 2015
    Co-Authors: Vincent X. Genest, Jean-michel Lemay, Luc Vinet, Alexei Zhedanov
    Abstract:

    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.

  • Two-variable $-1$ Jacobi Polynomials
    arXiv: Classical Analysis and ODEs, 2014
    Co-Authors: Vincent X. Genest, Jean-michel Lemay, Luc Vinet, Alexei Zhedanov
    Abstract:

    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 Wo\'zny. An alternative derivation of the weight function using Pearson-type equations is presented.

  • A LIMIT q = −1 FOR THE BIG Q-Jacobi Polynomials
    Transactions of the American Mathematical Society, 2012
    Co-Authors: Luc Vinet, Alexei Zhedanov
    Abstract:

    We study a new family of "classical" orthogonal Polynomials, here called big -1 Jacobi Polynomials, which satisfy (apart from a 3-term recur- rence relation) an eigenvalue problem with differential operators of Dunkl-type. These Polynomials can be obtained from the big q-Jacobi Polynomials in the limit q ! 1. An explicit expression of these Polynomials in terms of Gauss' hypergeometric functions is found. The big -1 Jacobi Polynomials are orthogo- nal on the union of two symmetric intervals of the real axis. We show that the big -1 Jacobi Polynomials can be obtained from the Bannai-Ito Polynomials when the orthogonality support is extended to an infinite number of points. We further indicate that these Polynomials provide a nontrivial realization of the Askey-Wilson algebra for q ! 1. The novelty lies in the fact that L is a differential-difference operator of special type. Namely, L is a linear operator which is of first order in the derivative operator @x and contains also the reflection operator R which acts as Rf(x) = f(−x). Roughly speaking, one can say that L belongs to the class of Dunkl operators (8) which contain both the operators @x and R. Nevertheless, the operator L differs from the standard Dunkl operators in a fundamental way. Indeed, L preserves the linear space of Polynomials of any given maximal degree. This basic property allows to construct a complete system of Polynomials Pn(x); n= 0;1;2;:::as eigenfunctions of the operator L. Guided by the q ! − 1 limit of the little q-Jacobi Polynomials, we derived in (20) an explicit expression of the Polynomials Pn(x) in terms of Gauss' hypergeometric functions. We also found explicitly the recurrence coefficients and s that the Polynomials Pn(x) are orthogonal on the interval (−1;1) with a weight function related to the weight function of the generalized Jacobi Polynomials (7). We also proved that they admit the Dunkl classical property (5) and further demonstrated that the operator L together with the multiplication operator x form a special case of the Askey-Wilson algebra AW(3) (21) corresponding to the parameter q = −1.

  • a limit q 1 for the big q Jacobi Polynomials
    Transactions of the American Mathematical Society, 2012
    Co-Authors: Luc Vinet, Alexei Zhedanov
    Abstract:

    We study a new family of "classical" orthogonal Polynomials, here called big -1 Jacobi Polynomials, which satisfy (apart from a 3-term recur- rence relation) an eigenvalue problem with differential operators of Dunkl-type. These Polynomials can be obtained from the big q-Jacobi Polynomials in the limit q ! 1. An explicit expression of these Polynomials in terms of Gauss' hypergeometric functions is found. The big -1 Jacobi Polynomials are orthogo- nal on the union of two symmetric intervals of the real axis. We show that the big -1 Jacobi Polynomials can be obtained from the Bannai-Ito Polynomials when the orthogonality support is extended to an infinite number of points. We further indicate that these Polynomials provide a nontrivial realization of the Askey-Wilson algebra for q ! 1. The novelty lies in the fact that L is a differential-difference operator of special type. Namely, L is a linear operator which is of first order in the derivative operator @x and contains also the reflection operator R which acts as Rf(x) = f(−x). Roughly speaking, one can say that L belongs to the class of Dunkl operators (8) which contain both the operators @x and R. Nevertheless, the operator L differs from the standard Dunkl operators in a fundamental way. Indeed, L preserves the linear space of Polynomials of any given maximal degree. This basic property allows to construct a complete system of Polynomials Pn(x); n= 0;1;2;:::as eigenfunctions of the operator L. Guided by the q ! − 1 limit of the little q-Jacobi Polynomials, we derived in (20) an explicit expression of the Polynomials Pn(x) in terms of Gauss' hypergeometric functions. We also found explicitly the recurrence coefficients and s that the Polynomials Pn(x) are orthogonal on the interval (−1;1) with a weight function related to the weight function of the generalized Jacobi Polynomials (7). We also proved that they admit the Dunkl classical property (5) and further demonstrated that the operator L together with the multiplication operator x form a special case of the Askey-Wilson algebra AW(3) (21) corresponding to the parameter q = −1.

  • -1 Krall-Jacobi Polynomials
    arXiv: Classical Analysis and ODEs, 2012
    Co-Authors: Luc Vinet, Alexei Zhedanov
    Abstract:

    We study a family of orthogonal Polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue equation involving a third order differential operator of Dunkl-type. The orthogonality measure of these Polynomials consists in the continuous measure of the little -1 Jacobi Polynomials to which is added an arbitrary mass located at the point $x=0$, the middle of the orthogonality interval. This provides the first nontrivial example of Krall-type Polynomials with a point mass inside the orthogonality interval. These Polynomials can be obtained by a Geronimus transform of the little $q$-Jacobi Polynomials in the limit $q=-1$.

Mumtaz Ahmad Khan - One of the best experts on this subject based on the ideXlab platform.

Michael Loss - One of the best experts on this subject based on the ideXlab platform.

  • on the markov sequence problem for Jacobi Polynomials
    Advances in Mathematics, 2011
    Co-Authors: Eric A. Carlen, Jeffrey S. Geronimo, Michael Loss
    Abstract:

    Abstract We give a simple and entirely elementary proof of Gasper's Theorem on the Markov sequence problem for Jacobi Polynomials. It is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding molecules introduced by Marc Kac, and the methods developed here yield new estimates relevant to the collision model.

  • On the Markov sequence problem for Jacobi Polynomials
    arXiv: Classical Analysis and ODEs, 2008
    Co-Authors: Eric A. Carlen, Jeffrey S. Geronimo, Michael Loss
    Abstract:

    We give a simple and entirely elementary proof of Gasper's theorem on the Markov sequence problem for Jacobi Polynomials. It is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding molecules introduced by Marc Kac. In the process, we obtain some new integral formulas for ratios of Jacobi Polynomials that generalize Gasper's product formula and a well known formula of Koornwinder.

Abdul Hakim Khan - One of the best experts on this subject based on the ideXlab platform.

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