Q derivative

Definition

The q-derivative of a function f(x) is defined as

It is also often written as D q f ( x ) . 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 g ( x ) = c x k . Then

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

Relationship to ordinary derivatives

Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the monomial is:

where [ n ] q is the q-bracket of n. Note that lim q → 1 [ n ] q = n 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, ( q ; q ) n is the q-Pochhammer symbol, and [ n ] q ! is the q-factorial. If f ( x ) is analytic we can apply the Taylor formula to the definition of D q ( f ( x ) ) to get

A q-analog of the Taylor expansion of a function about zero follows: