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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tsallis Statistics
The term Tsallis statistics usually refers to the collection of mathematical functions and associated probability distributions that were originated by Constantino Tsallis. Using that collection, it is possible to derive Tsallis distributions from the optimization of the Tsallis entropic form. A continuous real parameter ''q'' can be used to adjust the distributions, so that distributions which have properties intermediate to that of Gaussian and Lévy distributions can be created. The parameter ''q'' represents the degree of non- extensivity of the distribution. Tsallis statistics are useful for characterising complex, anomalous diffusion. Tsallis functions The ''q''-deformed exponential and logarithmic functions were first introduced in Tsallis statistics in 1994. However, the ''q''-deformation is the Box–Cox transformation for q=1-\lambda, proposed by George Box and David Cox in 1964. ''q''-exponential The ''q''-exponential is a deformation of the exponential function using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-derivative
In mathematics, in the area of combinatorics and quantum calculus, the ''q''-derivative, or Jackson derivative, is a ''q''-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's ''q''-integration. For other forms of q-derivative, see . Definition The ''q''-derivative of a function ''f''(''x'') is defined as :\left(\frac\right)_q f(x)=\frac. It is also often written as D_qf(x). The ''q''-derivative is also known as the Jackson derivative. Formally, in terms of Lagrange's shift operator in logarithmic variables, it amounts to the operator :D_q= \frac ~ \frac ~, which goes to the plain derivative \to \frac as q \to 1. It is manifestly linear, :\displaystyle D_q (f(x)+g(x)) = D_q f(x) + D_q g(x)~. It has a product rule analogous to the ordinary derivative product rule, with two equivalent forms :\displaystyle D_q (f(x)g(x)) = g(x)D_q f(x) + f(qx)D_q g(x) = g(qx)D_q f(x) + f(x)D_q g(x). Similarly, it satisfies a quotient rule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Dilogarithm
In mathematics, the quantum dilogarithm is a special function defined by the formula : \phi(x)\equiv(x;q)_\infty=\prod_^\infty (1-xq^n),\quad , q, 0. References * * * * * * * External links * {{nlab, id=quantum+dilogarithm, title=quantum dilogarithm Special functions Q-analogs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basic Hypergeometric Series
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 hypergeometric if the ratio of successive terms ''x''''n''+1/''x''''n'' is a rational function of ''n''. If the ratio of successive terms is a rational function of ''q''''n'', then the series is called a basic hypergeometric series. The number ''q'' is called the base. The basic hypergeometric series _2\phi_1(q^,q^;q^;q,x) was first considered by . It becomes the hypergeometric series F(\alpha,\beta;\gamma;x) in the limit when base q =1. Definition There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ. The unilateral basic hypergeometric series is defined as :\;_\phi_k \left begin a_1 & a_2 & \ldots & a_ \\ b_1 & b_2 & \ldots ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-analogs
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 chaoti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as Df = \lambda f for some scalar eigenvalue \lambda. The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvector. Eigenfunctions In general, an eigenvector of a linear operator ''D'' defined on some vector space is a nonzero vector in the domain of ''D'' that, when ''D'' acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where ''D'' is defined on a function space, the eigenvectors are referred to as eigenfunctions. That is, a function ''f'' is an eigenfunction of ''D'' if it satisfies the equation where λ is a scalar. The solutions to Equation may a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics". The exponential function satisfies the exponentiation identity e^ = e^x e^y \text x,y\in\mathbb, which, along with the definition e = \exp(1), shows that e^n=\underbrace_ for posi ... [...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 theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-bracket
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 theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
