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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, D_q \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 r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative (generalizations)
In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. Fréchet derivative The Fréchet derivative defines the derivative for general normed vector spaces V, W. Briefly, a function f : U \to W, where U is an open subset of V, is called ''Fréchet differentiable'' at x \in U if there exists a bounded linear operator A:V\to W such that \lim_ \frac = 0. Functions are defined as being differentiable in some open neighbourhood of x, rather than at individual points, as not doing so tends to lead to many pathological counterexamples. The Fréchet derivative is quite similar to the formula for the derivative found in elementary one-variable calculus, \lim_\frac = A, and simply moves ''A'' to the left hand side. However, the Fréchet derivative ''A'' denotes the function t \mapsto f'(x) \cdot t. In multivariable calculus, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-exponential
The term ''q''-exponential occurs in two contexts. The q-exponential distribution, based on the Tsallis q-exponential is discussed in elsewhere. 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. The ''q''-exponential is also known as the quantum dilogarithm. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-difference Polynomial
In combinatorial mathematics, the ''q''-difference polynomials or ''q''-harmonic polynomials are a polynomial sequence defined in terms of the ''q''-derivative. They are a generalized type of Brenke polynomial, and generalize the Appell polynomials. See also Sheffer sequence. Definition The q-difference polynomials satisfy the relation :\left(\frac \right)_q p_n(z) = \frac = \frac p_(z)= qp_(z) where the derivative symbol on the left is the q-derivative. In the limit of q\to 1, this becomes the definition of the Appell polynomials: :\fracp_n(z) = np_(z). Generating function The generalized generating function for these polynomials is of the type of generating function for Brenke polynomials, namely :A(w)e_q(zw) = \sum_^\infty \frac w^n where e_q(t) is the q-exponential: :e_q(t)=\sum_^\infty \frac= \sum_^\infty \frac. Here, q! is the q-factorial and :(q;q)_n=(1-q^n)(1-q^)\cdots (1-q) is the q-Pochhammer symbol In the mathematical field of combinatorics, the ''q''-P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Calculus
Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. The two types of calculus in quantum calculus are ''q''-calculus and ''h''-calculus. The goal of both types is to find "analogs" of mathematical objects, where, after taking a certain limit, the original object is returned. In ''q''-calculus, the limit as ''q'' tends to 1 is taken of the ''q''-analog. Likewise, in ''h''-calculus, the limit as h tends to 0 is taken of the ''h''-analog. The parameters q and h can be related by the formula q = e^h. Differentiation The ''q''-differential and ''h''-differential are defined as: :d_q(f(x)) = f(qx) - f(x) and :d_h(f(x)) = f(x + h) - f(x), respectively. The ''q''-derivative and ''h''-derivative are then defined as :D_q(f(x)) = \frac = \frac and :D_h(f(x)) = \frac = \frac respectively. By taking the limit as q \rightarrow 1 of the ''q''-derivative or as h \rightarrow 0 of the ''h''-derivati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |