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Wilson Polynomials
In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the Pochhammer symbol In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...s by :p_n(t^2)=(a+b)_n(a+c)_n(a+d)_n _4F_3\left( \begin -n&a+b+c+d+n-1&a-t&a+t \\ a+b&a+c&a+d \end ;1\right). See also * Askey–Wilson polynomials are a q-analogue of Wilson polynomials. References * *{{eom, id=Wilson_polynomials, title=Wilson polynomials, first=T.H. , last=Koornwinder Hypergeometric functions 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 orthogonality, 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 Pafnuty Chebyshev, P. L. Chebyshev and was pursued by Andrey Markov, A. A. Markov and Thomas Joannes Stieltjes, T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis (Gaussian quadrature, quadrature rules), probability theory, representation theory (of Lie group, Lie groups, quantum group, quantum groups, and re ... [...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, classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval [-1,1]. The Gegenbauer polynomials, and thus also the Legendre polynomials, Legendre, Zernike polynomials, 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 symbol, Pochhammer's symbol (for the rising factorial). In this case, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hahn Polynomials
In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 and rediscovered by Wolfgang Hahn . The Hahn class is a name for special cases of Hahn polynomials, including Hahn polynomials, Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials. Sometimes the Hahn class is taken to include limiting cases of these polynomials, in which case it also includes the classical orthogonal polynomials. Hahn polynomials are defined in terms of generalized hypergeometric functions by :Q_n(x;\alpha,\beta,N)= _3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).\ give a detailed list of their properties. If \alpha = \beta = 0, these polynomials are identical to the discrete Chebyshev polynomials except for a scale factor. Closely related polynomials include the dual Hahn polynomials ''R''''n''(''x'';γ,δ,''N''), the continuous Hahn polynomials ''p''''n''(''x'',' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charlier Polynomials
In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier. They are given in terms of the generalized hypergeometric function by :C_n(x; \mu)= _2F_0(-n,-x;-;-1/\mu)=(-1)^n n! L_n^\left(-\frac 1 \mu \right), where L are generalized Laguerre polynomials. They satisfy the orthogonality relation :\sum_^\infty \frac C_n(x; \mu)C_m(x; \mu)=\mu^ e^\mu n! \delta_, \quad \mu>0. They form a Sheffer sequence related to the Poisson process, similar to how Hermite polynomials relate to the Brownian motion. See also * Wilson polynomials In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the ..., a generalization of Charlier polynomials. References * C. V. L. Charlier (1905–1906) ''Über die Darstellung willk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Hypergeometric Function
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. Notation A hypergeometric series is formally defined as a power series :\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_ \beta_n z^n in which the ratio of successive coefficients is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pochhammer Symbol
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \end The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as :\begin x^ = x^\overline &= \overbrace^ \\ &= \prod_^n(x+k-1) = \prod_^(x+k) \,. \end The value of each is taken to be 1 (an empty product) when . These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation , where is a non-negative integer. It may represent ''either'' the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used with yet another meaning, namely to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |