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Jacobi Polynomials
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of Classical orthogonal polynomials, classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval [-1,1]. The Gegenbauer polynomials, and thus also the Legendre polynomials, Legendre, Zernike polynomials, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. The definition is in IV.1; the differential equation – in IV.2; Rodrigues' formula is in IV.3; the generating function is in IV.4; the recurrent relation is in IV.5. The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi. Definitions Via the hypergeometric function The Jacobi polynomials are defined via the hypergeometric function as follows: :P_n^(z)=\frac\,_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac(1-z)\right), where (\alpha+1)_n is Pochhammer symbol, Pochhammer's symbol (for the rising factorial). In this case, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plot Of The Jacobi Polynomial Function P N^(a,b) With N=10 And A=2 And B=2 In The Complex Plane From -2-2i To 2+2i With Colors Created With Mathematica 13
Plot or Plotting may refer to: Art, media and entertainment * Plot (narrative), the story of a piece of fiction Music * The Plot (album), ''The Plot'' (album), a 1976 album by jazz trumpeter Enrico Rava * The Plot (band), a band formed in 2003 Other * Plot (film), ''Plot'' (film), a 1973 French-Italian film * Plotting (video game), ''Plotting'' (video game), a 1989 Taito puzzle video game, also called Flipull * The Plot (video game), ''The Plot'' (video game), a platform game released in 1988 for the Amstrad CPC and Sinclair Spectrum * Plotting (non-fiction), ''Plotting'' (non-fiction), a 1939 book on writing by Jack Woodford * The Plot (novel), ''The Plot'' (novel), a 2021 mystery by Jean Hanff Korelitz Graphics * Plot (graphics), a graphical technique for representing a data set * Plot (radar), a graphic display that shows all collated data from a ship's on-board sensors * Plot plan, a type of drawing which shows existing and proposed conditions for a given area Land * Plot ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except ... [...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] |
Romanovski Polynomials
In mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski in French transcription) within the context of probability distribution functions in statistics. They form an orthogonal subset of a more general family of little-known Routh polynomials introduced by Edward John Routh in 1884. The term Romanovski polynomials was put forward by Raposo, with reference to the so-called 'pseudo-Jacobi polynomials in Lesky's classification scheme. It seems more consistent to refer to them as Romanovski–Routh polynomials, by analogy with the terms Romanovski–Bessel and Romanovski–Jacobi used by Lesky for two other sets of orthogonal polynomials. In some contrast to the standard classical orthogonal polynomials, the polynomials under consideration differ, in so far as for arbitrary parameters only ''a finite number of them are orthogonal'', as discussed in more detail below. The differential equatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Process
Jacobi may refer to: * People with the surname Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenvalue algorithm, a method for calculating the eigenvalues and eigenvectors of a real symmetric matrix * Jacobi elliptic functions, a set of doubly-periodic functions * Jacobi polynomials, a class of orthogonal polynomials * Jacobi symbol, a generalization of the Legendre symbol * Jacobi coordinates, a simplification of coordinates for an n-body system * Jacobi identity for non-associative binary operations * Jacobi's formula for the derivative of the determinant of a matrix * Jacobi triple product an identity in the theory of theta functions * Jacobi's theorem (other) (various) Other: * Jacobi Medical Center, New York * Jacobi (grape), another name for the French/German wine grape Pinot Noir Précoce * Jacobi (crater), a lunar impact crater in the so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo Jacobi Polynomials
In mathematics, the term Pseudo Jacobi polynomials was introduced by Lesky for one of three finite sequences of orthogonal polynomials y. Since they form an orthogonal subset of Routh polynomials it seems consistent to refer to them as Romanovski-Routh polynomials, by analogy with the terms Romanovski-Bessel and Romanovski-Jacobi used by Lesky. As shown by Askey for two other sequencesth is finite sequence orthogonal polynomials of can be expressed in terms of Jacobi polynomials In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of Classical orthogonal polynomials, classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta ... of imaginary argument. In following Raposo et al. they are often referred to simply as Romanovski polynomials. References {{reflist Orthogonal polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Continuous Q-Jacobi Polynomials
In mathematics, the continuous ''q''-Jacobi polynomials ''P''(''x'', ''q''), introduced by , 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 and the q-Pochhammer symbol In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ... by :P_n^(x;q)=\frac_4\phi_3\left begin q^,q^,q^,q^\\ q^,-q^,-q^\end ;q,q\rightqquad x=\cos\,\theta. References * * * * * *{{cite thesis , last=Sadjang , first=Patrick Njionou , title=Moments of Classical Orthogonal Polynomials , type=Ph.D. , publisher=Universität Kassel , citeseerx=10.1.1.643.3896 Orthogonal polynomials Q-analogs Special hypergeometric functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big Q-Jacobi Polynomials
In mathematics, the big ''q''-Jacobi polynomials ''P''''n''(''x'';''a'',''b'',''c'';''q''), introduced by , 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 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,c;q)=_3\phi_2(q^,abq^,x;aq,cq;q,q) References * * * * Orthogonal polynomials Q-analogs Special hypergeometric functions {{mathematics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Askey–Gasper Inequality
In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by and used in the proof of the Bieberbach conjecture. Statement It states that if \beta\geq 0, \alpha+\beta\geq -2, and -1\leq x\leq 1 then :\sum_^n \frac \ge 0 where :P_k^(x) is a Jacobi polynomial. The case when \beta=0 can also be written as :_3F_2 \left (-n,n+\alpha+2,\tfrac(\alpha+1);\tfrac(\alpha+3),\alpha+1;t \right)>0, \qquad 0\leq t-1. In this form, with a non-negative integer, the inequality was used by Louis de Branges in his proof of the Bieberbach conjecture. Proof gave a short proof of this inequality, by combining the identity :\begin \frac &\times _3F_2 \left (-n,n+\alpha+2,\tfrac(\alpha+1);\tfrac(\alpha+3),\alpha+1;t \right) = \\ &= \frac \times _3F_2\left (-n+2j,n-2j+\alpha+1,\tfrac(\alpha+1);\tfrac(\alpha+2),\alpha+1;t \right ) \end with the Clausen inequality. Generalizations give some generalizations of the Askey–Gasper inequality to basic hypergeometri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wigner D-matrix
The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The letter stands for ''Darstellung'', which means "representation" in German. Definition of the Wigner D-matrix Let be generators of the Lie algebra of SU(2) and SO(3). In quantum mechanics, these three operators are the components of a vector operator known as ''angular momentum''. Examples are the angular momentum of an electron in an atom, electronic spin, and the angular momentum of a rigid rotor. In all cases, the three operators satisfy the following commutation relations, : _x,J_y= i J_z,\quad _z,J_x= i J_y,\quad _y,J_z= i J_x, where ''i'' is the purely imaginary number and Planck's constant has been set equal to one. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain (mathematical Analysis)
In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space or the complex coordinate space . This is a different concept than the domain of a function, though it is often used for that purpose, for example in partial differential equations and Sobolev spaces. The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the term ''domain'', some use the term ''region'', some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as ''non-empty connected open subset''. One common convention is to define a ''domain'' as a connected open se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
