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Chihara Polynomials (other)
In mathematics, Chihara polynomials may refer to one of the families of orthogonal polynomials studied by Theodore Seio Chihara, including *Al-Salam–Chihara polynomials *Brenke–Chihara polynomials In mathematics, Brenke polynomials are special cases of generalized Appell polynomials, and Brenke–Chihara polynomials are the Brenke polynomials that are also orthogonal polynomials. introduced sequences of Brenke polynomials ''P'n'', whi ... * Chihara–Ismail polynomials {{mathdab ... [...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] |
Theodore Seio Chihara
Theodore Seio Chihara (born 1929) is a mathematician working on orthogonal polynomials who introduced Al-Salam–Chihara polynomials, Brenke–Chihara polynomials, and Chihara–Ismail polynomials. His brother is composer Paul Chihara Paul Seiko Chihara (born July 9, 1938) is an American composer. Life and career Chihara was born in Seattle, Washington in 1938. A Japanese American, he spent three years of his childhood with his family in an internment camp in Minidoka, Idah .... Publications * * References * * {{DEFAULTSORT:Chihara, Theodore Seio Living people 20th-century American mathematicians 21st-century American mathematicians 1929 births ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Al-Salam–Chihara Polynomials
In mathematics, the Al-Salam–Chihara polynomials ''Q''''n''(''x'';''a'',''b'';''q'') are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of the properties of Al-Salam–Chihara polynomials. Definition The Al-Salam–Chihara polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ... by : Q_n(x;a,b;q) = \frac_3\phi_2(q^, ae^, ae^; ab,0; q,q) where ''x'' = cos(θ). References * * * * Further reading * Bryc, W., Matysiak, W., & Szabłowski, P. (2005). Probabilistic aspects of Al-Salam–Chihara polynomials. Proceedings of the American Mathematical Society, 133(4), 1127-1134. * Floreanini, R., LeTourneux, J., & Vinet, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brenke–Chihara Polynomials
In mathematics, Brenke polynomials are special cases of generalized Appell polynomials, and Brenke–Chihara polynomials are the Brenke polynomials that are also orthogonal polynomials. introduced sequences of Brenke polynomials ''P''''n'', which are special cases of generalized Appell polynomials with generating function of the form :A(w)B(xw)=\sum_^\infty P_n(x)w^n. Brenke observed that Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ... and Laguerre polynomials are examples of Brenke polynomials, and asked if there are any other sequences of orthogonal polynomials of this form. found some further examples of orthogonal Brenke polynomials. completely classified all Brenke polynomials that form orthogonal sequences, which are now called Brenke–Chihara pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |