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Bateman Polynomials
In mathematics, the Bateman polynomials are a family ''F''''n'' of orthogonal polynomials introduced by . The Bateman–Pasternack polynomials are a generalization introduced by . Bateman polynomials can be defined by the relation :F_n\left(\frac\right)\operatorname(x) = \operatorname(x)P_n(\tanh(x)). where ''P''''n'' is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by :F_n(x)=_3F_2\left(\begin-n,~n+1,~\tfrac12(x+1)\\ 1,~1 \end; 1\right). generalized the Bateman polynomials to polynomials ''F'' with :F_n^m\left(\frac\right)\operatorname^(x) = \operatorname^(x)P_n(\tanh(x)) These generalized polynomials also have a representation in terms of generalized hypergeometric functions, namely :F_n^m(x)=_3F_2\left(\begin-n,~n+1,~\tfrac12(x+m+1)\\ 1,~m+1 \end; 1\right). showed that the polynomials ''Q''''n'' studied by , see Touchard polynomials, are the same as Bateman polynomials up to a change of variable: more precisely : Q_n(x)=(-1)^n2^nn! ... [...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] |
Legendre Polynomial
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions. Definition by construction as an orthogonal system In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w(x) = 1 over the interval 1,1/math>. That is, P_n(x) is a polynomial of degree n, such that \int_^1 P_m(x) P_n(x) \,dx = 0 \quad \text n \ne m. With the additional standardization con ... [...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] |
Touchard Polynomials
The Touchard polynomials, studied by , also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by :T_n(x)=\sum_^n S(n,k)x^k=\sum_^n \left\x^k, where S(n,k)=\left\is a Stirling number of the second kind, i.e., the number of partitions of a set of size ''n'' into ''k'' disjoint non-empty subsets. Properties Basic properties The value at 1 of the ''n''th Touchard polynomial is the ''n''th Bell number, i.e., the number of partitions of a set of size ''n'': :T_n(1)=B_n. If ''X'' is a random variable with a Poisson distribution with expected value λ, then its ''n''th moment is E(''X''''n'') = ''T''''n''(λ), leading to the definition: :T_(x)=e^\sum_^\infty \frac . Using this fact one can quickly prove that this polynomial sequence is of binomial type, i.e., it satisfies the sequence of identities: :T_n(\lambda+\mu)=\sum_^n T_k(\lambda) T_(\mu). The Touchard polynomials constitute the only polynomial sequence o ... [...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] |
Canadian Journal Of Mathematics
The ''Canadian Journal of Mathematics'' (french: Journal canadien de mathématiques) is a bimonthly mathematics journal published by the Canadian Mathematical Society. It was established in 1949 by H. S. M. Coxeter and G. de B. Robinson. The current editors-in-chief of the journal are Louigi Addario-Berry and Eyal Goren. The journal publishes articles in all areas of mathematics. See also * Canadian Mathematical Bulletin The ''Canadian Mathematical Bulletin'' (french: Bulletin Canadien de Mathématiques) is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antoni ... References External links * University of Toronto Press academic journals Mathematics journals Publications established in 1949 Bimonthly journals Multilingual journals Cambridge University Press academic journals Academic journals associated with learned and professional societies of Canada ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * |
