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List Of Q-analogs
{{DISPLAYTITLE:List of ''q''-analogs This is a list of q-analog, ''q''-analogs in mathematics and related fields. Algebra * Iwahori–Hecke algebra * Quantum affine algebra * Quantum enveloping algebra * Quantum group Analysis * Jackson integral * q-derivative, ''q''-derivative * Q-difference polynomial, ''q''-difference polynomial * Quantum calculus Combinatorics * LLT polynomial * Gaussian binomial coefficient, ''q''-binomial coefficient * q-Pochhammer symbol, ''q''-Pochhammer symbol * q-Vandermonde identity, ''q''-Vandermonde identity Orthogonal polynomials * q-Bessel polynomials, ''q''-Bessel polynomials * q-Charlier polynomials, ''q''-Charlier polynomials * q-Hahn polynomials, ''q''-Hahn polynomials * ''q''-Jacobi polynomials: **Big q-Jacobi polynomials, Big ''q''-Jacobi polynomials **Continuous q-Jacobi polynomials, Continuous ''q''-Jacobi polynomials **Little q-Jacobi polynomials, Little ''q''-Jacobi polynomials * q-Krawtchouk polynomials, ''q''-Krawtchouk polyno ... [...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 chaotic ... [...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] |
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] |
Tsallis Entropy
In physics, the Tsallis entropy is a generalization of the standard Boltzmann–Gibbs entropy. Overview The concept was introduced in 1988 by Constantino Tsallis as a basis for generalizing the standard statistical mechanics and is identical in form to Havrda–Charvát structural α-entropy, introduced in 1967 within information theory. In scientific literature, the physical relevance of the Tsallis entropy has been debated. However, from the years 2000 on, an increasingly wide spectrum of natural, artificial and social complex systems have been identified which confirm the predictions and consequences that are derived from this nonadditive entropy, such as nonextensive statistical mechanics, which generalizes the Boltzmann–Gibbs theory. Among the various experimental verifications and applications presently available in the literature, the following ones deserve a special mention: # The distribution characterizing the motion of cold atoms in dissipative optical lattices pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Gaussian Distribution
The ''q''-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. It is one example of a Tsallis distribution. The ''q''-Gaussian is a generalization of the Gaussian in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. The normal distribution is recovered as ''q'' → 1. The ''q''-Gaussian has been applied to problems in the fields of statistical mechanics, geology, anatomy, astronomy, economics, finance, and machine learning. The distribution is often favored for its heavy tails in comparison to the Gaussian for 1 < ''q'' < 3. For the ''q''-Gaussian distribution is the PDF of a bounded . This makes in biology and other domains the ''q''-Gaussian distribution more suitable than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Weibull Distribution
In statistics, the ''q''-Weibull distribution is a probability distribution that generalizes the Weibull distribution and the Lomax distribution (Pareto Type II). It is one example of a Tsallis distribution. Characterization Probability density function The probability density function of a ''q''-Weibull random variable is: : f(x;q,\lambda,\kappa) = \begin (2-q)\frac\left(\frac\right)^ e_q(-(x/\lambda)^)& x\geq0 ,\\ 0 & x 0 is the ''scale parameter'' of the distribution and :e_q(x) = \begin \exp(x) & \textq=1, \\ pt +(1-q)x & \textq \ne 1 \text 1+(1-q)x >0, \\ pt0^ & \textq \ne 1\text1+(1-q)x \le 0, \\ pt\end is the ''q''-exponential Cumulative distribution function The cumulative distribution function of a ''q''-Weibull random variable is: :\begin1- e_^ & x\geq0\\ 0 & x<0\end where : : Mean The mean of the ''q''-Weibull distribution is : |
Q-exponential Distribution
The ''q''-exponential distribution is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints, including constraining the domain to be positive. It is one example of a Tsallis distribution. The ''q''-exponential is a generalization of the exponential distribution in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. The exponential distribution is recovered as q \rightarrow 1. Originally proposed by the statisticians George Box and David Cox in 1964, and known as the reverse Box–Cox transformation for q=1-\lambda, a particular case of power transform in statistics. Characterization Probability density function The ''q''-exponential distribution has the probability density function :(2-q) \lambda e_q(-\lambda x) where :e_q(x) = +(1-q)x is the ''q''-exponential if . When , ''e''''q''(x) is just exp(''x''). Derivation In a similar procedure to how the exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Q-distribution
In mathematical physics and probability and statistics, the Gaussian ''q''-distribution is a family of probability distributions that includes, as limiting cases, the uniform distribution and the normal (Gaussian) distribution. It was introduced by Diaz and Teruel, is a q-analog of the Gaussian or normal distribution. The distribution is symmetric about zero and is bounded, except for the limiting case of the normal distribution. The limiting uniform distribution is on the range -1 to +1. Definition Let ''q'' be a real number in the interval , 1). The probability density function of the Gaussian ''q''-distribution is given by :s_q(x) = \begin 0 & \text x \nu. \end where :\nu = \nu(q) = \frac , : c(q)=2(1-q)^\sum_^\infty \frac . The ''q''-analogue [''t'']''q'' of the real number t is given by : [t]_q=\frac. The ''q''-analogue of the exponential function is the q-exponential, ''E'', which is given by : E_q^=\sum_^q^\frac where the ''q''-analogue of the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Racah Polynomials
In mathematics, the ''q''-Racah polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric functions 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 ... by :p_n(q^+q^cd;a,b,c,d;q) = _4\phi_3\left begin q^ &abq^&q^&q^cd\\ aq&bdq&cq\\ \end;q;q\right/math> They are sometimes given with changes of variables as :W_n(x;a,b,c,N;q) = _4\phi_3\left begin q^ &abq^&q^&cq^\\ aq&bcq&q^\\ \end;q;q\right/math> Relation to other polynomials q-Racah polynomials→Racah polynomials References * * * *{{dlmf, id=18, title=Chapter 18: Orthogonal Polynomials, first=Tom H. , last=Koornwinder, first2=Roderick ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Meixner–Pollaczek Polynomials
In mathematics, the ''q''-Meixner–Pollaczek polynomials 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 ：Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analoques, p 460,Springer : P_(x;a\mid q) = a^ e^ \frac_3\phi_2(q^, ae^, ae^; a^2, 0 \mid q; q),\quad x=\cos(\theta+\phi). References * * * {{DEFAULTSORT:Q-Meixner-Pollaczek polynomials Orthogonal polynomials Q-analogs Special hypergeometric functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Meixner Polynomials
In mathematics, the ''q''-Meixner polynomials 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 :M_n(q^;b,c;q)=_2\phi_1\left begin q^,q^\\ bq\end ;q,-\frac\right 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] |
Q-Laguerre Polynomials
In mathematics, the ''q''-Laguerre polynomials, or generalized Stieltjes–Wigert polynomials ''P''(''x'';''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 ''q''-Laguerre 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 hy ...s and the q-Pochhammer symbol by :\displaystyle L_n^(x;q) = \frac _1\phi_1(q^;q^;q,-q^x). Orthogonality Orthogonality is defined by the unimono nature of the polynomials' convergence at boundaries in integral form. References * * * *{{citation , last=Moak, first=Daniel S., title=The q-analogue of the Laguerre polynomials, journal=J. Math. Anal. Appl., volume=81, issue=1, pages=20†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
