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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 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 ...s 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Waleed Al-Salam
Waleed Al-Salam (born 15 July 1926 in Baghdad, Iraq – died 14 April 1996 in Edmonton, Canada) was a mathematician who introduced Al-Salam–Chihara polynomials, Al-Salam–Carlitz polynomials, q-Konhauser polynomials, and Al-Salam–Ismail polynomials. He was a Professor Emeritus ''Emeritus/Emerita'' () is an honorary title granted to someone who retirement, retires from a position of distinction, most commonly an academic faculty position, but is allowed to continue using the previous title, as in "professor emeritus". ... at the University of Alberta. Born in Iraq, Baghdad, Al-Salam received his bachelor's degree in engineering physics (1950) and M.A. in mathematics (1951) from University of California, Berkeley, University of California Berkeley. He completed his education at Duke University, Duke, receiving his Doctor of Philosophy, Ph.D. for his dissertation on Bessel polynomials (1958). References * * External linksWaleed Al-Salam 1926-1996 * {{DEFAULTSORT ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Richard Askey
Richard Allen Askey (June 4, 1933 – October 9, 2019) was an American mathematician, known for his expertise in the area of special functions. The Askey–Wilson polynomials (introduced by him in 1984 together with James A. Wilson) are on the top level of the (q-)Askey scheme, which organizes orthogonal polynomials of (q-)hypergeometric type into a hierarchy. The Askey–Gasper inequality for Jacobi polynomials is essential in de Brange's famous proof of the Bieberbach conjecture. Biography Askey earned a B.A. at Washington University in St. Louis in 1955, an M.A. at Harvard University in 1956, and a Ph.D. at Princeton University in 1961. After working as an instructor at Washington University (1958–1961) and University of Chicago (1961–1963), he joined the faculty of the University of Wisconsin–Madison in 1963 as an Assistant Professor of Mathematics. He became a full professor at Wisconsin in 1968, and since 2003 was a professor emeritus. Askey was a Guggenheim Fel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 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 * * ..., and sieved Jacobi polynomials. References * Orthogonal polynomials {{polynomial-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Ultraspherical Polynomials
In mathematics, Gegenbauer polynomials or ultraspherical polynomials ''C''(''x'') are orthogonal polynomials on the interval minus;1,1with respect to the weight function (1 − ''x''2)''α''–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer. Characterizations File:Plot of the Gegenbauer polynomial C n^(m)(x) with n=10 and m=1 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg, Plot of the Gegenbauer polynomial C n^(m)(x) with n=10 and m=1 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D File:Mplwp gegenbauer Cn05a1.svg, Gegenbauer polynomials with ''α''=1 File:Mplwp gegenbauer Cn05a2.svg, Gegenbauer polynomials with ''α''=2 File:Mplwp gegenbauer Cn05a3.svg, Gegenbauer polynomials with ''α''=3 File:Gegenbauer polynomials.gif, An animation showing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. Its ISSN number is 0002-9947. 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'' References External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR ( ; short for ''Journal Storage'') is a digital library of academic journals, books, and primary sources founded in 1994. Originally containing digitized back issues of academic journals, it now encompasses books and other primary source ... American Mathematical Society academic journals Mathematics jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |