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Hahn Polynomial
In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 and rediscovered by Wolfgang Hahn . The Hahn class is a name for special cases of Hahn polynomials, including Hahn polynomials, Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials. Sometimes the Hahn class is taken to include limiting cases of these polynomials, in which case it also includes the classical orthogonal polynomials. Hahn polynomials are defined in terms of generalized hypergeometric functions by :Q_n(x;\alpha,\beta,N)= _3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).\ give a detailed list of their properties. If \alpha = \beta = 0, these polynomials are identical to the discrete Chebyshev polynomials except for a scale factor. Closely related polynomials include the dual Hahn polynomials ''R''''n''(''x'';γ,δ,''N''), the continuous Hahn polynomials ''p''''n''(''x'',' ... [...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] |
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] |
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] |
Racah Polynomials
In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients. The Racah polynomials were first defined by and are given by :p_n(x(x+\gamma+\delta+1)) = _4F_3\left begin -n &n+\alpha+\beta+1&-x&x+\gamma+\delta+1\\ \alpha+1&\gamma+1&\beta+\delta+1\\ \end;1\right Orthogonality :\sum_^N\operatorname_n(x;\alpha,\beta,\gamma,\delta) \operatorname_m(x;\alpha,\beta,\gamma,\delta)\frac \omega_y=h_n\operatorname_, :when \alpha+1=-N, :where \operatorname is the Racah polynomial, :x=y(y+\gamma+\delta+1), :\operatorname_ is the Kronecker delta function and the weight functions are :\omega_y=\frac, :and :h_n=\frac\frac\frac, :(\cdot)_n is the Pochhammer symbol. Rodrigues-type formula :\omega(x;\alpha,\beta,\gamma,\delta)\operatorname_n(\lambda(x);\alpha,\beta,\gamma,\delta)=(\gamma+\delta+1)_n\frac\omega(x;\alpha+n,\beta+n,\gamma+n,\delta), :where \nabla is the backwa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Hahn Polynomials
In mathematics, the ''q''-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric functions by :Q_n(q^;a,b,N;q)=_3\phi_2\left begin q^,abq^,q^\\ aq,q^\end ;q,q\right Relation to other polynomials q-Hahn polynomials→ Quantum q-Krawtchouk polynomials： \lim_Q_(q^;a;p,N, q)=K_^(q^;p,N;q) q-Hahn polynomials→ Hahn polynomials make the substitution\alpha=q^,\beta=q^ into definition of q-Hahn polynomials, and find the limit q→1, we obtain :_3F_2(-n,\alpha+\beta+n+1,-x,\alpha+1,-N,1)，which is exactly Hahn polynomials In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 and rediscovered by Wolfgang Hahn . The Hahn class is a name for spec .... References * * * *{{cite journal, la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Dual Hahn Polynomials
In mathematics, the continuous dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by :S_n(x^2;a,b,c)= _3F_2(-n,a+ix,a-ix;a+b,a+c;1).\ give a detailed list of their properties. Closely related polynomials include the dual Hahn polynomials ''R''''n''(''x'';γ,δ,''N''), the continuous Hahn polynomials ''p''''n''(''x'',''a'',''b'', , ), and the Hahn polynomials. These polynomials all have ''q''-analogs with an extra parameter ''q'', such as the q-Hahn polynomials ''Q''''n''(''x'';α,β, ''N'';''q''), and so on. Relation to other polynomials *Wilson polynomials In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the ... are a generalization of continuou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Hahn Polynomials
In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by :p_n(x;a,b,c,d)= i^n\frac_3F_2\left( \begin -n, n+a+b+c+d-1, a+ix \\ a+c, a+d \end ; 1\right) give a detailed list of their properties. Closely related polynomials include the dual Hahn polynomials ''R''''n''(''x'';γ,δ,''N''), the Hahn polynomials ''Q''''n''(''x'';''a'',''b'',''c''), and the continuous dual Hahn polynomials ''S''''n''(''x'';''a'',''b'',''c''). These polynomials all have ''q''-analogs with an extra parameter ''q'', such as the q-Hahn polynomials ''Q''''n''(''x'';α,β, ''N'';''q''), and so on. Orthogonality The continuous Hahn polynomials ''p''''n''(''x'';''a'',''b'',''c'',''d'') are orthogonal with respect to the weight function :w(x)=\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix). In particular, they satisfy the orthogonality relation : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Hahn Polynomials
In mathematics, the dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined on a non-uniform lattice x(s)=s(s+1) and are defined as :w_n^ (s,a,b)=\frac _3F_2(-n,a-s,a+s+1;a-b+a,a+c+1;1) for n=0,1,...,N-1 and the parameters a,b,c are restricted to -\frac Orthogonality The dual Hahn polynomials have the orthogonality condit ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Chebyshev Polynomials
In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev and rediscovered by Gram. They were later found to be applicable to various algebraic properties of spin angular momentum. Elementary Definition The discrete Chebyshev polynomial t^N_n(x) is a polynomial of degree ''n'' in ''x'', for n = 0, 1, 2,\ldots, N -1, constructed such that two polynomials of unequal degree are orthogonal with respect to the weight function w(x) = \sum_^ \delta(x-r), with \delta(\cdot) being the Dirac delta function. That is, \int_^ t^N_n(x) t^N_m (x) w(x) \, dx = 0 \quad \text \quad n \ne m . The integral on the left is actually a sum because of the delta function, and we have, \sum_^ t^N_n(r) t^N_m (r) = 0 \quad \text\quad n \ne m. Thus, even though t^N_n(x) is a polynomial in x, only its values at a discrete set of points, x = 0, 1, 2, \ldots, N-1 are of any significance. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Hypergeometric Function
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. Notation A hypergeometric series is formally defined as a power series :\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_ \beta_n z^n in which the ratio of successive coefficients is a ... [...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] |
Classical Orthogonal Polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials). They have many important applications in such areas as mathematical physics (in particular, the theory of random matrices), approximation theory, numerical analysis, and many others. Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien-Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, the study of continued fractions to solve the moment problem by P. L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials. For given polynomials Q, L: \R \to \R and \forall\,n \in \N_0 the classical orthogonal polynomials f_n:\R \to \R are characterized by being solutions of the differential equation :Q(x) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
