q-derivative

The q-derivative is also known as the Jackson derivative.

The n-th q-derivative of a function may be given as:

The q-derivative, the difference operator and the standard derivative can all be viewed as the same thing on different time scales.

The eigenfunction of the q-derivative is the q-exponential e(x).

For example, the q-derivative of the monomial is:

In combinatorial mathematics, the q-difference polynomials or q-harmonic polynomials are a polynomial sequence defined in terms of the q-derivative.

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.

For x nonzero, if f is a differentiable function of x then in the limit as q 1 we obtain the ordinary derivative, thus the q-derivative may be viewed as its q-deformation.

One may proceed further and develop, for example, equivalent notions of Taylor expansion, et cetera, and even arrive at q-calculus analogues for all of the usual functions one would want to have, such as an analogue for the sine function whose q-derivative is the appropriate analogue for the cosine.