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Pseudo Jacobi Polynomials
In mathematics, the term Pseudo Jacobi polynomials was introduced by Lesky for one of three finite sequences of orthogonal polynomials y. Since they form an orthogonal subset of Routh polynomials it seems consistent to refer to them as Romanovski-Routh polynomials, by analogy with the terms Romanovski-Bessel and Romanovski-Jacobi used by Lesky. As shown by Askey for two other sequencesth is finite sequence orthogonal polynomials of can be expressed in terms of Jacobi polynomials In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval 1,1/math>. The ... of imaginary argument. In following Raposo et al. they are often referred to simply as Romanovski polynomials. References {{reflist Orthogonal polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P. L. Chebyshev and was pursued by A. A. Markov and T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis ( quadrature rules), probability theory, representation theory (of Lie groups, quantum groups, and related objects), enumerative combinatorics, algebraic combinatorics, mathematical physics (the theory of r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Polynomials
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval 1,1/math>. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. The definition is in IV.1; the differential equation – in IV.2; Rodrigues' formula is in IV.3; the generating function is in IV.4; the recurrent relation is in IV.5. The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi. Definitions Via the hypergeometric function The Jacobi polynomials are defined via the hypergeometric function as follows: :P_n^(z)=\frac\,_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac(1-z)\right), where (\alpha+1)_n is Pochhammer's symbol (for the rising factorial). In this case, the series for the hypergeometric function is finite, therefore one obtains the follow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |