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| In [[mathematics]], in the area of [[combinatorics]], 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 integral|Jackson's q-integration]]
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| ==Definition==
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| The q-derivative of a function ''f''(''x'') is defined as
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| :<math>\left(\frac{d}{dx}\right)_q f(x)=\frac{f(qx)-f(x)}{qx-x}.</math>
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| It is also often written as <math>D_qf(x)</math>. The q-derivative is also known as the '''Jackson derivative'''.
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| Formally, in terms of Lagrange's [[shift operator]] in logarithmic variables, it amounts to the operator
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| :<math>D_q= \frac{1}{x} ~ \frac{q^{d~~~ \over d (\ln x)} -1}{q-1} ~, </math>
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| which goes to the plain derivative, →<sup>''d''</sup>⁄<sub>''dx''</sub>, as ''q''→1.
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| It is manifestly linear,
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| :<math>\displaystyle D_q (f(x)+g(x)) = D_q f(x) + D_q g(x)~.</math>
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| It has product rule analogous to the ordinary derivative product rule, with two equivalent forms
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| :<math>\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). </math>
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| Similarly, it satisfies a quotient rule,
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| :<math>\displaystyle D_q (f(x)/g(x)) = \frac{g(x)D_q f(x) - f(x)D_q g(x)}{g(qx)g(x)},\quad g(x)g(qx)\neq 0. </math>
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| There is also a rule similar to the chain rule for ordinary derivatives. Let <math>g(x) = c x^k</math>. Then
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| :<math>\displaystyle D_q f(g(x)) = D_{q^k}(f)(g(x))D_q(g)(x).</math>
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| The [[eigenfunction]] of the q-derivative is the [[q-exponential]] ''e<sub>q</sub>''(''x'').
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| ==Relationship to ordinary derivatives==
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| Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the [[monomial]] is:
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| :<math>\left(\frac{d}{dz}\right)_q z^n = \frac{1-q^n}{1-q} z^{n-1} =
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| [n]_q z^{n-1}</math>
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| where <math>[n]_q</math> is the [[q-bracket]] of ''n''. Note that <math>\lim_{q\to 1}[n]_q = n</math> so the ordinary derivative is regained in this limit.
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| The ''n''-th q-derivative of a function may be given as:
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| :<math>(D^n_q f)(0)=
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| \frac{f^{(n)}(0)}{n!} \frac{(q;q)_n}{(1-q)^n}=
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| \frac{f^{(n)}(0)}{n!} [n]_q!
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| </math>
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| provided that the ordinary ''n''-th derivative of ''f'' exists at ''x''=0. Here, <math>(q;q)_n</math> is the [[q-Pochhammer symbol]], and <math>[n]_q!</math> is the [[q-factorial]]. If <math>f(x)</math> is analytic we can apply the Taylor formula to the definition of <math>D_q(f(x)) </math> to get
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| :<math>\displaystyle D_q(f(x)) = \sum_{k=0}^{\infty}\frac{(q-1)^k}{(k+1)!} x^k f^{(k+1)}(x).</math>
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| A q-analog of the Taylor expansion of a function about zero follows:
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| :<math>f(z)=\sum_{n=0}^\infty f^{(n)}(0)\,\frac{z^n}{n!} = \sum_{n=0}^\infty (D^n_q f)(0)\,\frac{z^n}{[n]_q!}</math>
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| == See also ==
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| * [[Derivative (generalizations)]]
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| * [[Jackson integral]]
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| * [[Q-exponential]]
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| * [[Q-difference polynomial]]s
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| * [[Quantum calculus]]
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| * [[Tsallis entropy]]
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| ==References==
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| * F. H. Jackson (1908), "On q-functions and a certain difference operator", ''Trans. Roy. Soc. Edin.'', '''46''' 253-281.
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|
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| * Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538
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| * Victor Kac, Pokman Cheung, ''Quantum Calculus'', Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8
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| ==Further reading==
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| * J. Koekoek, R. Koekoek, ''[http://arxiv.org/abs/math/9908140 A note on the q-derivative operator]'', (1999) ArXiv math/9908140
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| * Thomas Ernst, ''[http://www.math.uu.se/research/pub/Ernst4.pdf The History of q-Calculus and a new method]'',(2001),
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| [[Category:Q-analogs]]
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| [[Category:Differential calculus]]
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| [[Category:Linear operators in calculus]]
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