|
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] |
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] |
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] |
Binomial Type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers \left\ in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities :p_n(x+y)=\sum_^n\, p_k(x)\, p_(y). Many such sequences exist. The set of all such sequences forms a Lie group under the operation of umbral composition, explained below. Every sequence of binomial type may be expressed in terms of the Bell polynomials. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type). Polynomial sequences put on firm footing the vague 19th century notions of umbral calculus. Examples * In consequence of this definition the binomial theorem can be stated by saying that the sequence is of binomial type. * The sequence of " lower factorials" is defined by(x)_n=x(x-1)(x-2)\cdot\cdots\cdot(x-n+1).(In the theory of special functions, this same notation denotes upper fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Appell Sequence
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence \_ satisfying the identity :\frac p_n(x) = np_(x), and in which p_0(x) is a non-zero constant. Among the most notable Appell sequences besides the trivial example \ are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments. Equivalent characterizations of Appell sequences The following conditions on polynomial sequences can easily be seen to be equivalent: * For n = 1, 2, 3,\ldots, ::\frac p_n(x) = n p_(x) :and p_0(x) is a non-zero constant; * For some sequence \_^ of scalars with c_0 \neq 0, ::p_n(x) = \sum_^n \binom c_k x^; * For the same sequence of scalars, ::p_n(x) = \left(\sum_^\infty \frac D^k\right) x^n, :where ::D = \frac; * For n=0,1,2,\ldots, ::p_n(x+y) = \sum_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rodrigues Formula
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by , and . The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out in 1865 that Rodrigues was the first to discover it. The term is also used to describe similar formulas for other orthogonal polynomials. describes the history of the Rodrigues formula in detail. Statement Let \_^\infty be a sequence of orthogonal polynomials satisfying the orthogonality condition \int_a^b P_m(x) P_n(x) w(x) \, dx = K_n \delta_, where w(x) is a suitable weight function, K_n is a constant depending on n, and \delta_ is the Kronecker delta. If the weight function w(x) satisfies the following differential equation (called Pearson's differential equation), \frac = \frac, where A(x) is a polynomial with degree at most 1 and B(x) is a polynomial with degree at most 2 and, further, the limits \lim_ w(x) B(x) = 0, \qquad \lim_ w( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of a wave function, the quantum-mechanical characterization of an isolated physical system. The equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Equation
Chebyshev's equation is the second order linear differential equation : (1-x^2) - x + p^2 y = 0 where p is a real (or complex) constant. The equation is named after Russian mathematician Pafnuty Chebyshev. The solutions can be obtained by power series: :y = \sum_^\infty a_nx^n where the coefficients obey the recurrence relation : a_ = a_n. The series converges for , x, <1 (note, ''x'' may be complex), as may be seen by applying the to the recurrence. The recurrence may be started with arbitrary values of ''a''0 and ''a''1, leading to the two-dimensional space of solutions that arises from second order differential equations. The standard choices are: :''a''0 = 1 ; ''a''1 = 0, leading to the solution : and :''a''0 = 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associated Legendre Polynomials
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation \left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, or equivalently \frac \left \left(1 - x^2\right) \frac P_\ell^m(x) \right+ \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, where the indices ''ℓ'' and ''m'' (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on only if ''ℓ'' and ''m'' are integers with 0 ≤ ''m'' ≤ ''ℓ'', or with trivially equivalent negative values. When in addition ''m'' is even, the function is a polynomial. When ''m'' is zero and ''ℓ'' integer, these functions are identical to the Legendre polynomials. In general, when ''ℓ'' and ''m'' are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Olinde Rodrigues
Benjamin Olinde Rodrigues (6 October 1795 – 17 December 1851), more commonly known as Olinde Rodrigues, was a French banker, mathematician, and social reformer. In mathematics Rodrigues is remembered for Rodrigues' rotation formula for vectors, the Rodrigues formula about series of orthogonal polynomials and the Euler–Rodrigues parameters. Biography Rodrigues was born into a well-to-do Sephardi Jewish family in Bordeaux. He was awarded a doctorate in mathematics on 28 June 1815 by the University of Paris. His dissertation contains the result now called Rodrigues' formula. After graduation, Rodrigues became a banker. A close associate of the Comte de Saint-Simon, Rodrigues continued, after Saint-Simon's death in 1825, to champion the older man's socialist ideals, a school of thought that came to be known as Saint-Simonianism. During this period, Rodrigues published writings on politics, social reform, and banking. In 1840 he published a result on transformation groups, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rodrigues' Formula
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by , and . The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out in 1865 that Rodrigues was the first to discover it. The term is also used to describe similar formulas for other orthogonal polynomials. describes the history of the Rodrigues formula in detail. Statement Let \_^\infty be a sequence of orthogonal polynomials satisfying the orthogonality condition \int_a^b P_m(x) P_n(x) w(x) \, dx = K_n \delta_, where w(x) is a suitable weight function, K_n is a constant depending on n, and \delta_ is the Kronecker delta. If the weight function w(x) satisfies the following differential equation (called Pearson's differential equation), \frac = \frac, where A(x) is a polynomial with degree at most 1 and B(x) is a polynomial with degree at most 2 and, further, the limits \lim_ w(x) B(x) = 0, \qquad \lim_ w( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.png "Click for more on -> Schrödinger Equation")
