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Meixner–Pollaczek Polynomials
In mathematics, the Meixner–Pollaczek polynomials are a family of orthogonal polynomials ''P''(''x'',φ) introduced by , which up to elementary changes of variables are the same as the Pollaczek polynomials ''P''(''x'',''a'',''b'') rediscovered by in the case λ=1/2, and later generalized by him. They are defined by :P_n^(x;\phi) = \frace^_2F_1\left(\begin -n,~\lambda+ix\\ 2\lambda \end; 1-e^\right) :P_n^(\cos \phi;a,b) = \frace^_2F_1\left(\begin-n,~\lambda+i(a\cos \phi+b)/\sin \phi\\ 2\lambda \end;1-e^\right) Examples The first few Meixner–Pollaczek polynomials are :P_0^(x;\phi)=1 :P_1^(x;\phi)=2(\lambda\cos\phi + x\sin\phi) :P_2^(x;\phi)=x^2+\lambda^2+(\lambda^2+\lambda-x^2)\cos(2\phi)+(1+2\lambda)x\sin(2\phi). Properties Orthogonality The Meixner–Pollaczek polynomials ''P''m(λ)(''x'';φ) are orthogonal on the real line with respect to the weight function : w(x; \lambda, \phi)= , \Gamma(\lambda+ix), ^2 e^ and the orthogonality relation is given by :\int_^P_n^(x;\p ... [...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] |
Sieved Pollaczek Polynomials
In mathematics, sieved Pollaczek polynomials are a family of sieved orthogonal polynomials In mathematics, sieved orthogonal polynomials are orthogonal polynomials whose recurrence relations are formed by sieving the recurrence relations of another family; in other words, some of the recurrence relations are replaced by simpler ones. Th ..., introduced by . Their recurrence relations are a modified (or "sieved") version of the recurrence relations for Pollaczek polynomials. References * * *{{Citation , last1=Ismail , first1=Mourad E. H. , title=On sieved orthogonal polynomials. I. Symmetric Pollaczek analogues , doi=10.1137/0516081 , mr=800799 , year=1985 , journal=SIAM Journal on Mathematical Analysis , issn=0036-1410 , volume=16 , issue=5 , pages=1093–1113 Orthogonal polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |