|
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. The first examples were the sieved ultraspherical polynomials introduced by . Mourad Ismail later studied sieved orthogonal polynomials in a long series of papers. Other families of sieved orthogonal polynomials that have been studied include sieved Pollaczek polynomials, and sieved Jacobi polynomials. References *{{Citation , last1=Al-Salam , first1=Waleed , last2=Allaway , first2=W. R. , last3=Askey , first3=Richard , title=Sieved ultraspherical polynomials , doi=10.2307/1999273 , mr=742411 , year=1984 , journal=Transactions of the American Mathematical Society , issn=0002-9947 , volume=284 , issue=1 , pages=39–55, jstor=1999273 , citeseerx=10.1.1.308.3668 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] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sieved Ultraspherical Polynomials
In mathematics, the two families ''c''(''x'';''k'') and ''B''(''x'';''k'') of sieved ultraspherical polynomials, introduced by Waleed Al-Salam, W.R. Allaway and Richard Askey in 1984, are the archetypal examples of sieved orthogonal polynomials. Their recurrence relations are a modified (or "sieved") version of the recurrence relations for ultraspherical polynomials. Recurrence relations For the sieved ultraspherical polynomials of the first kind the recurrence relations are :2xc_n^\lambda(x;k) = c_^\lambda(x;k) + c_^\lambda(x;k) if ''n'' is not divisible by ''k'' :2x(m+\lambda)c_^\lambda(x;k) = (m+2\lambda)c_^\lambda(x;k) + mc_^\lambda(x;k) For the sieved ultraspherical polynomials of the second kind the recurrence relations are :2xB_^\lambda(x;k) = B_^\lambda(x;k) + B_^\lambda(x;k) if ''n'' is not divisible by ''k'' :2x(m+\lambda)B_^\lambda(x;k) = mB_^\lambda(x;k) +(m+2\lambda)B_^\lambda(x;k) References *{{Citation , last1=Al-Salam , first1=Waleed , last2=Allaway , first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. See also * ''Bulletin of the American Mathematical Society'' * '' Journal of the American Mathematical Society'' * ''Memoirs of the American Mathematical Society'' * ''Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... American Mathematical Society academic journals Mathematics journals Publications established in 1900 {{math-journal-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mourad Ismail
Mourad E. H. Ismail (born April 27, 1944, in Cairo, Egypt) is a mathematician working on orthogonal polynomials and special functions. Ismail received his bachelor's degree from Cairo University. He holds Masters and doctorate degrees from the University of Alberta. He worked at and visited several universities. Currently he holds a research professorship at the University of Central Florida and a Distinguished Scientist Fellowship at King Saud University in Saudi Arabia. Ismail is a fellow of the American Mathematical Society retrieved 2014-12-17 and the . He is among the ISI highly cited scientists. He s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sieved Pollaczek Polynomials
In mathematics, sieved Pollaczek polynomials are a family of sieved orthogonal polynomials, 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] |
Sieved Jacobi Polynomials
In mathematics, sieved Jacobi polynomials are a family of sieved orthogonal polynomials, introduced by . Their recurrence relations are a modified (or "sieved") version of the recurrence relations for Jacobi polynomials. References * *{{Citation , last1=Askey , first1=Richard , editor1-last=Jackson , editor1-first=David M. , editor2-last=Vanstone , editor2-first=Scott A. , title=Enumeration and design (Waterloo, Ont., 1982) , chapter-url=https://books.google.com/books?id=Pg-FAAAAIAAJ , publisher=Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference ... , location=Boston, MA , isbn=978-0-12-379120-7 , mr=782309 , year=1984 , chapter=Orthogonal polynomials old and new, and some combinatorial connections , page67–84, url-access=registration , url=https://a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |