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Little Q-Jacobi Polynomials
In mathematics, the little ''q''-Jacobi polynomials ''p''''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 their properties. Definition The little ''q''-Jacobi polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called h ...s by :\displaystyle p_n(x;a,b;q) = _2\phi_1(q^,abq^;aq;q,xq) Gallery The following are a set of animation plots for Little ''q''-Jacobi polynomials, with varying q; three density plots of imaginary, real and modulus in complex space; three set of complex 3D plots of imaginary, real and modulus of the said polynomials. References * * * *{{dlmf, id=18, first=Tom H. , last=Koornwinder, first2=Rod ... [...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] |
Askey Scheme
In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in , the Askey scheme was first drawn by and by , and has since been extended by and to cover basic orthogonal polynomials. Askey scheme for hypergeometric orthogonal polynomials give the following version of the Askey scheme: ;_4F_3(4): Wilson , Racah ;_3F_2(3): Continuous dual Hahn , Continuous Hahn , Hahn , dual Hahn ;_2F_1(2): Meixner–Pollaczek , Jacobi , Pseudo Jacobi , Meixner , Krawtchouk ;_2F_0(1)\ \ / \ \ _1F_1(1): Laguerre , Bessel , Charlier ;_2F_0(0): Hermite Here _pF_q(n) indicates a hypergeometric series representation with n parameters Askey scheme for basic hypergeometric orthogonal polynomials give the following scheme for basic hypergeometric orthogonal polynomials: ;4\phi3: Askey–Wilson , q-Racah ;3\phi2: Continuous dual q-Hahn , ... [...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 internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basic Hypergeometric Function
In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x''''n'' is called hypergeometric if the ratio of successive terms ''x''''n''+1/''x''''n'' is a rational function of ''n''. If the ratio of successive terms is a rational function of ''q''''n'', then the series is called a basic hypergeometric series. The number ''q'' is called the base. The basic hypergeometric series _2\phi_1(q^,q^;q^;q,x) was first considered by . It becomes the hypergeometric series F(\alpha,\beta;\gamma;x) in the limit when base q =1. Definition There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ. The unilateral basic hypergeometric series is defined as :\;_\phi_k \left begin a_1 & a_2 & \ldots & a_ \\ b_1 & b_2 & \ldots & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LITTLE Q-JACOBI POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT
Little is a synonym for small size and may refer to: Arts and entertainment * ''Little'' (album), 1990 debut album of Vic Chesnutt * ''Little'' (film), 2019 American comedy film *The Littles, a series of children's novels by American author John Peterson ** ''The Littles'' (TV series), an American animated series based on the novels Places *Little, Kentucky, United States *Little, West Virginia, United States Other uses *Clan Little, a Scottish clan *Little (surname), an English surname *Little (automobile), an American automobile manufactured from 1912 to 1915 *Little, Brown and Company, an American publishing company * USS ''Little'', multiple United States Navy ships See also * * *Little Mountain (other) * Little River (other) *Little Island (other) Little Island can refer to: Geographical areas Australia * Little Island (South Australia) * Little Island (Tasmania) * Little Island (Western Australia) Canada * Little Island (Lake Kagawong), Ontario ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LITTLE Q-JACOBI POLYNOMIALS IM COMPLEX 3D MAPLE PLOT
Little is a synonym for small size and may refer to: Arts and entertainment * ''Little'' (album), 1990 debut album of Vic Chesnutt * ''Little'' (film), 2019 American comedy film *The Littles, a series of children's novels by American author John Peterson ** ''The Littles'' (TV series), an American animated series based on the novels Places *Little, Kentucky, United States *Little, West Virginia, United States Other uses *Clan Little, a Scottish clan *Little (surname), an English surname *Little (automobile), an American automobile manufactured from 1912 to 1915 *Little, Brown and Company, an American publishing company * USS ''Little'', multiple United States Navy ships See also * * *Little Mountain (other) * Little River (other) *Little Island (other) Little Island can refer to: Geographical areas Australia * Little Island (South Australia) * Little Island (Tasmania) * Little Island (Western Australia) Canada * Little Island (Lake Kagawong), Ontario ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LITTLE Q-JACOBI POLYNOMIALS RE COMPLEX 3D MAPLE PLOT
Little is a synonym for small size and may refer to: Arts and entertainment * ''Little'' (album), 1990 debut album of Vic Chesnutt * ''Little'' (film), 2019 American comedy film *The Littles, a series of children's novels by American author John Peterson ** ''The Littles'' (TV series), an American animated series based on the novels Places *Little, Kentucky, United States *Little, West Virginia, United States Other uses *Clan Little, a Scottish clan *Little (surname), an English surname *Little (automobile), an American automobile manufactured from 1912 to 1915 *Little, Brown and Company, an American publishing company * USS ''Little'', multiple United States Navy ships See also * * *Little Mountain (other) * Little River (other) *Little Island (other) Little Island can refer to: Geographical areas Australia * Little Island (South Australia) * Little Island (Tasmania) * Little Island (Western Australia) Canada * Little Island (Lake Kagawong), Ontario ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LITTLE Q-JACOBI POLYNOMIALS ABS DENSITY MAPLE PLOT
Little is a synonym for small size and may refer to: Arts and entertainment * ''Little'' (album), 1990 debut album of Vic Chesnutt * ''Little'' (film), 2019 American comedy film *The Littles, a series of children's novels by American author John Peterson ** ''The Littles'' (TV series), an American animated series based on the novels Places *Little, Kentucky, United States *Little, West Virginia, United States Other uses *Clan Little, a Scottish clan *Little (surname), an English surname *Little (automobile), an American automobile manufactured from 1912 to 1915 *Little, Brown and Company, an American publishing company * USS ''Little'', multiple United States Navy ships See also * * *Little Mountain (other) * Little River (other) *Little Island (other) Little Island can refer to: Geographical areas Australia * Little Island (South Australia) * Little Island (Tasmania) * Little Island (Western Australia) Canada * Little Island (Lake Kagawong), Ontario ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LITTLE Q-JACOBI POLYNOMIALS IM DENSITY MAPLE PLOT
Little is a synonym for small size and may refer to: Arts and entertainment * ''Little'' (album), 1990 debut album of Vic Chesnutt * ''Little'' (film), 2019 American comedy film *The Littles, a series of children's novels by American author John Peterson ** ''The Littles'' (TV series), an American animated series based on the novels Places *Little, Kentucky, United States *Little, West Virginia, United States Other uses *Clan Little, a Scottish clan *Little (surname), an English surname *Little (automobile), an American automobile manufactured from 1912 to 1915 *Little, Brown and Company, an American publishing company * USS ''Little'', multiple United States Navy ships See also * * *Little Mountain (other) * Little River (other) *Little Island (other) Little Island can refer to: Geographical areas Australia * Little Island (South Australia) * Little Island (Tasmania) * Little Island (Western Australia) Canada * Little Island (Lake Kagawong), Ontario ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LITTLE Q-JACOBI POLYNOMIALS RE DENSITY MAPLE PLOT
Little is a synonym for small size and may refer to: Arts and entertainment * ''Little'' (album), 1990 debut album of Vic Chesnutt * ''Little'' (film), 2019 American comedy film *The Littles, a series of children's novels by American author John Peterson ** ''The Littles'' (TV series), an American animated series based on the novels Places *Little, Kentucky, United States *Little, West Virginia, United States Other uses *Clan Little, a Scottish clan *Little (surname), an English surname *Little (automobile), an American automobile manufactured from 1912 to 1915 *Little, Brown and Company, an American publishing company * USS ''Little'', multiple United States Navy ships See also * * *Little Mountain (other) * Little River (other) *Little Island (other) Little Island can refer to: Geographical areas Australia * Little Island (South Australia) * Little Island (Tasmania) * Little Island (Western Australia) Canada * Little Island (Lake Kagawong), Ontario ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Nachrichten
''Mathematische Nachrichten'' (abbreviated ''Math. Nachr.''; English: ''Mathematical News'') is a mathematical journal published in 12 issues per year by Wiley-VCH GmbH. It should not be confused with the ''Internationale Mathematische Nachrichten'', an unrelated publication of the Austrian Mathematical Society. It was established in 1948 by East German mathematician Erhard Schmidt, who became its first editor-in-chief. At that time it was associated with the German Academy of Sciences at Berlin, and published by Akademie Verlag. After the fall of the Berlin Wall, Akademie Verlag was sold to VCH Verlagsgruppe Weinheim, which in turn was sold to John Wiley & Sons. According to the 2020 edition of Journal Citation Reports, the journal had an impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
