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Discrete Q-Hermite Polynomials
In mathematics, the discrete ''q''-Hermite polynomials are two closely related families ''h''''n''(''x'';''q'') and ''ĥ''''n''(''x'';''q'') of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. ''h''''n''(''x'';''q'') is also called discrete q-Hermite I polynomials and ''ĥ''''n''(''x'';''q'') is also called discrete q-Hermite II polynomials. Definition The discrete ''q''-Hermite polynomials are given in terms of basic hypergeometric functions and the Al-Salam–Carlitz polynomials In mathematics, Al-Salam–Carlitz polynomials ''U''(''x'';''q'') and ''V''(''x'';''q'') are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. Definit ... by :\displaystyle h_n(x;q)=q^_2\phi_1(q^,x^;0;q,-qx) = x^n_2\phi_0(q^,q^;;q^2,q^/x^2) = U_n^(x;q) :\displaystyle \hat h_n(x;q)=i^q^_2\phi_0(q^,ix;;q,-q^n) = x^n_2\phi_ ... [...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] |