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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 continu ... [...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 orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ... 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. These are frequently given by the Rodrigues' formula. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and wa ... [...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-Hah ... [...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 orthogonal polynomials, 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 succe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Dual Hahn 1
Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous game, a generalization of games used in game theory ** Law of continuity, a heuristic principle of Gottfried Leibniz * Continuous function, in particular: ** Continuity (topology), a generalization to functions between topological spaces ** Scott continuity, for functions between posets ** Continuity (set theory), for functions between ordinals ** Continuity (category theory), for functors ** Graph continuity, for payoff functions in game theory * Continuity theorem may refer to one of two results: ** Lévy's continuity theorem, on random variables ** Kolmogorov continuity theorem, on stochastic processes * In geometry: ** Parametric continuity, for parametrised curves ** Geometric continuity, a concept primarily applied to the conic sectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Plot Of Continuous Duall Hahn Polynomials
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * ''Complex'' (album), by Montaigne, 2019, and its title track * ''Complex'' (EP), by Rifle Sport, 1985 * "Complex" (song), by Gary Numan, 1979 * "Complex", a song by Katie Gregson-MacLeod, 2022 * "Complex" a song by Be'O and Zico, 2022 * Complex Networks, publisher of the now-only-online magazine ''Complex'' Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-protein ... [...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, ... [...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] |
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] |
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 special cases of Hahn polynomials, including Hahn polynomials, Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials. Sometimes the Hahn class is taken to include limiting case (mathematics), 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 po ... [...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 spe .... References * * * *{{cite journal, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Pochhammer symbol In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial \begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) . \end ...s by :p_n(t^2)=(a+b)_n(a+c)_n(a+d)_n _4F_3\left( \begin -n&a+b+c+d+n-1&a-t&a+t \\ a+b&a+c&a+d \end ;1\right). See also * Askey–Wilson polynomials are a q-analogue of Wilson polynomials. References * * Hypergeometric functions Orthogonal polynomials {{polynomial-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Hypergeometric Functions
Special or specials may refer to: Policing * Specials, Ulster Special Constabulary, the Northern Ireland police force * Specials, Special Constable, an auxiliary, volunteer, or temporary; police worker or police officer * Special police forces Military * Special forces * Special operations Literature * ''Specials'' (novel), a novel by Scott Westerfeld * ''Specials'', the comic book heroes, see ''Rising Stars'' (comic) Film and television * Special (lighting), a stage light that is used for a single, specific purpose * ''Special'' (film), a 2006 scifi dramedy * ''The Specials'' (2000 film), a comedy film about a group of superheroes * Special 26, a 2013 Indian Hindi-language period heist thriller film * ''The Specials'' (2019 film), a film by Olivier Nakache and Éric Toledano * Television special, television programming that temporarily replaces scheduled programming * ''Special'' (TV series), a 2019 Netflix Original TV series * ''Specials'' (TV series), a 1991 TV series ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
