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's q-integration.
The q-derivative of a function f(x) is defined as
It is also often written as . 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
which goes to the plain derivative, → d⁄dx, as q → 1.
It is manifestly linear,
It has product rule analogous to the ordinary derivative product rule, with two equivalent forms
Similarly, it satisfies a quotient rule,
There is also a rule similar to the chain rule for ordinary derivatives. Let . Then
The eigenfunction of the q-derivative is the q-exponential eq(x).
Relationship to ordinary derivatives
Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the monomial is:
where is the q-bracket of n. Note that so the ordinary derivative is regained in this limit.
The n-th q-derivative of a function may be given as:
provided that the ordinary n-th derivative of f exists at x = 0. Here, is the q-Pochhammer symbol, and is the q-factorial. If is analytic we can apply the Taylor formula to the definition of to get
A q-analog of the Taylor expansion of a function about zero follows:
- F. H. Jackson (1908), On q-functions and a certain difference operator, Trans. Roy. Soc. Edin., 46 253-281.
- Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538
- Victor Kac, Pokman Cheung, Quantum Calculus, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8