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Charlier Polynomials
In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier. They are given in terms of the generalized hypergeometric function by :C_n(x; \mu)= _2F_0(-n,-x;-;-1/\mu)=(-1)^n n! L_n^\left(-\frac 1 \mu \right), where L are generalized Laguerre polynomials. They satisfy the orthogonality relation :\sum_^\infty \frac C_n(x; \mu)C_m(x; \mu)=\mu^ e^\mu n! \delta_, \quad \mu>0. They form a Sheffer sequence related to the Poisson process, similar to how Hermite polynomials relate to the Brownian motion. See also * 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 ..., a generalization of Charlier polynomials. References * C. V. L. Charlier (1905–1906) ''Über die Darstellung willk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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] |
Carl Charlier
Carl Vilhelm Ludwig Charlier (1 April 1862 – 4 November 1934) was a Swedish astronomer. His parents were Emmerich Emanuel and Aurora Kristina (née Hollstein) Charlier. Career Charlier was born in Östersund. He received his Ph.D. from Uppsala University in 1887, later worked there and at the Stockholm Observatory and was Professor of Astronomy and Director of the Observatory at Lund University from 1897. He made extensive statistical studies of the stars in our galaxy and their positions and motions, and tried to develop a model of the galaxy based on this. He proposed the siriometer as a unit of stellar distance. Charlier was also interested in pure statistics and played a role in the development of statistics in Swedish academia. Several of his pupils became statisticians, working at universities and in government and companies. Related to his work on galactic structure, he also developed a cosmological theory based on the work of Johann Heinrich Lambert. In the re ... [...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] |
Laguerre Polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only if is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of xy'' + (\alpha + 1 - x)y' + ny = 0~. where is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin). More generally, a Laguerre function is a solution when is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form \int_0^\infty f(x) e^ \, dx. These polynomials, usually denoted , , …, are a polynomial sequence which may be defined by the Rodrigues formula, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonality Relation
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations. Applications Characters of irreducible representations encode many important prop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheffer Sequence
In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are named for Isador M. Sheffer. Definition Fix a polynomial sequence (''p''''n''). Define a linear operator ''Q'' on polynomials in ''x'' by :Qp_n(x) = np_(x)\, . This determines ''Q'' on all polynomials. The polynomial sequence ''p''''n'' is a ''Sheffer sequence'' if the linear operator ''Q'' just defined is ''shift-equivariant''; such a ''Q'' is then a delta operator. Here, we define a linear operator ''Q'' on polynomials to be ''shift-equivariant'' if, whenever ''f''(''x'') = ''g''(''x'' + ''a'') = ''T''''a'' ''g''(''x'') is a "shift" of ''g''(''x''), then (''Qf'')(''x'') = (''Qg'')(''x'' + ''a''); i.e., ''Q'' commutes with every shift operator: ''T''''a''''Q'' = ''QT''''a''. Properties The set of all Sheffer sequences is a group un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field. This point process has convenient mathematical properties, which has led to its being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50(3):311–326, 1996. biology,H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. ''Journal of mathematical biology'', 26(3):263–298, 1988. ecology,H. Thompson. Spatial point processes, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as in connection with Brownian motion; * combinatorics, as an example of an Appell sequence, obeying the umbral calculus; * numerical analysis as Gaussian quadrature; * physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term \beginxu_\end is present); * systems theory in connection with nonlinear operations on Gaussian noise. * random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem). This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking throu ... [...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) \,. \e ...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 * *{{eom, id=Wilson_polynomials, title=Wilson polynomials, first=T.H. , last=Koornwinder Hypergeometric functions Orthogonal polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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