q-derivative
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In mathematics, in the area of combinatorics, the q-derivative is a q-analog of the ordinary derivative.
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[edit] Definition
The q-derivative of a function f(x) is defined as
It is also often written as Dqf(x). The q-derivative is also known as the Jackson derivative.
[edit] Relationship to ordinary derivatives
Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the monomial is:
![\left(\frac{d}{dz}\right)_q z^n = \frac{1-q^n}{1-q} z^{n-1} =
[n]_q z^{n-1}](../../../../math/9/9/4/9947b7d31a33f0f4adad24722f9ac84d.png)
where [n]q is the q-bracket of n. Note that ![\lim_{q\to 1}[n]_q = n](../../../../math/a/1/6/a16ea7eae0dca1369c416d8a9e3ee329.png)
so the ordinary derivative is regained in this limit.
The n 'th derivative of a function may be given as
![(D^n_q f)(0)=
\frac{f^{(n)}(0)}{n!} \frac{(q;q)_n}{(1-q)^n}=
\frac{f^{(n)}(0)}{n!} [n]_q!](../../../../math/9/8/6/986d4f5262809d95412717ade3e57f4a.png)
provided that the ordinary n 'th derivative of f exists at x=0. Here, (q;q)n is the q-Pochhammer symbol, and [n]q! is the q-factorial.
[edit] See also
- Derivative (generalizations)
- Jackson integral
- Q-exponential
- Q-difference polynomials
- Quantum calculus
[edit] References
- Victor Kac, Pokman Cheung, Quantum Calculus, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8
- J. Koekoek, R. Koekoek, A note on the q-derivative operator, (1999) ArXiv math/9908140
