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

\left(\frac{d}{dx}\right)_q f(x)=\frac{f(qx)-f(x)}{qx-x}

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}

where [n]q is the q-bracket of n. Note that \lim_{q\to 1}[n]_q = n 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!

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

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