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Q-gamma Function
In q-analog theory, the q-gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by . It is given by \Gamma_q(x) = (1-q)^\prod_^\infty \frac=(1-q)^\,\frac when , q, 1. Here (\cdot;\cdot)_\infty is the infinite q-Pochhammer symbol. The q-gamma function satisfies the functional equation \Gamma_q(x+1) = \frac\Gamma_q(x)= q\Gamma_q(x) In addition, the q-gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey (). For non-negative integers ''n'', \Gamma_q(n)= -1q! where cdotq is the q-factorial function. Thus the q-gamma function can be considered as an extension of the q-factorial function to the real numbers. The relation to the ordinary gamma function is made explicit in the limit \lim_ \Gamma_q(x) = \Gamma(x). There is a simple proof of this limit by Gosper. See the appendix of (). Transformation properties The q-gamma function satis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-analog
In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q''-analogs that arise naturally, rather than in arbitrarily contriving ''q''-analogs of known results. The earliest ''q''-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, , , ''q''-analogues are most frequently studied in the mathematical fields of combinatorics and special functions. In these settings, the limit is often formal, as is often discrete-valued (for example, it may represent a prime power). ''q''-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Gamma Function
In mathematics, the multiple gamma function \Gamma_N is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was studied by . At the end of this paper he mentioned the existence of multiple gamma functions generalizing it, and studied these further in . Double gamma functions \Gamma_2 are closely related to the q-gamma function, and triple gamma functions \Gamma_3 are related to the elliptic gamma function. Definition For \Re a_i>0, let :\Gamma_N(w\mid a_1,\ldots,a_N) = \exp\left(\left.\frac \zeta_N(s,w \mid a_1, \ldots, a_N) \_ \right)\ , where \zeta_N is the Barnes zeta function. (This differs by a constant from Barnes's original definition.) Properties Considered as a meromorphic function of w, \Gamma_N(w\mid a_1,\ldots,a_N) has no zeros. It has poles at w= -\sum_^N n_ia_i for non-negative integers n_i. These poles are simple unless some of them coincide. Up to multiplication by the exponential of a polynomial, \Gamma_N(w\mid a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 symbol (x)_n = x(x+1)\dots(x+n-1), in the sense that \lim_ \frac = (x)_n. The ''q''-Pochhammer symbol is a major building block in the construction of ''q''-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike the ordinary Pochhammer symbol, the ''q''-Pochhammer symbol can be extended to an infinite product: (a;q)_\infty = \prod_^ (1-aq^k). This is an analytic function of ''q'' in the interior of the unit disk, and can also be considered as a formal power series in ''q''. The special case \phi(q) = (q;q)_\infty=\prod_^\infty (1-q^k) is known as Euler's function, and is important in combinatorics, number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bohr–Mollerup Theorem
In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for by :\Gamma(x)=\int_0^\infty t^ e^\,dt as the ''only'' positive function , with domain on the interval , that simultaneously has the following three properties: * , and * for and * is logarithmically convex. A treatment of this theorem is in Artin's book ''The Gamma Function'', which has been reprinted by the AMS in a collection of Artin's writings. The theorem was first published in a textbook on complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ..., as Bohr and Mollerup thought it had already been proved. Statement :Bohr–Mollerup Theorem. is the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Askey
Richard Allen Askey (4 June 1933 – 9 October 2019) was an American mathematician, known for his expertise in the area of special functions. The Askey–Wilson polynomials (introduced by him in 1984 together with James A. Wilson) are on the top level of the (q-)Askey scheme, which organizes orthogonal polynomials of (q-)hypergeometric type into a hierarchy. The Askey–Gasper inequality for Jacobi polynomials is essential in de Brange's famous proof of the Bieberbach conjecture. Askey earned a B.A. at Washington University in 1955, an M.A. at Harvard University in 1956, and a Ph.D. at Princeton University in 1961. After working as an instructor at Washington University (1958–1961) and University of Chicago (1961–1963), he joined the faculty of the University of Wisconsin–Madison in 1963 as an Assistant Professor of Mathematics. He became a full professor at Wisconsin in 1968, and since 2003 was a professor emeritus. Askey was a Guggenheim Fellow, 1969–1970, which acad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-factorial
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 symbol (x)_n = x(x+1)\dots(x+n-1), in the sense that \lim_ \frac = (x)_n. The ''q''-Pochhammer symbol is a major building block in the construction of ''q''-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike the ordinary Pochhammer symbol, the ''q''-Pochhammer symbol can be extended to an infinite product: (a;q)_\infty = \prod_^ (1-aq^k). This is an analytic function of ''q'' in the interior of the unit disk, and can also be considered as a formal power series in ''q''. The special case \phi(q) = (q;q)_\infty=\prod_^\infty (1-q^k) is known as Euler's function, and is important in combinatorics, number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heaviside Step Function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one. The function was originally developed in operational calculus for the solution of differential equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely. Oliver Heaviside, who developed the operational calculus as a tool in the analysis of telegraphic communications, represented the function as . The Heaviside function may be defined as: * a piecewise function: H(x) := \begin 1, & x > 0 \\ 0, & x \le 0 \end * using the Iverson bracket notation: H(x) := 0.html" ;"title=">0">>0/math> * an indicator function: H(x) := \mathbf_=\mathbf 1_(x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive-definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-exponential
In combinatorial mathematics, a ''q''-exponential is a ''q''-analog of the exponential function, namely the eigenfunction of a ''q''-derivative. There are many ''q''-derivatives, for example, the classical ''q''-derivative, the Askey-Wilson operator, etc. Therefore, unlike the classical exponentials, ''q''-exponentials are not unique. For example, e_q(z) is the ''q''-exponential corresponding to the classical ''q''-derivative while \mathcal_q(z) are eigenfunctions of the Askey-Wilson operators. Definition The ''q''-exponential e_q(z) is defined as :e_q(z)= \sum_^\infty \frac = \sum_^\infty \frac = \sum_^\infty z^n\frac where _q is the ''q''-factorial and :(q;q)_n=(1-q^n)(1-q^)\cdots (1-q) is the ''q''-Pochhammer symbol. That this is the ''q''-analog of the exponential follows from the property :\left(\frac\right)_q e_q(z) = e_q(z) where the derivative on the left is the ''q''-derivative. The above is easily verified by considering the ''q''-derivative of the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
