Q-derivative

In mathematics, in the area of combinatorics and quantum calculus, 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. For other forms of q-derivative, see Chung et al. (1994).

Definition

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, as .

It is manifestly linear,

It has a 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 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 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:

Higher order q-derivatives

The following representation for higher order -derivatives is known:

is the -binomial coefficient. By changing the order of summation as , we obtain the next formula:

Higher order -derivatives are used to -Taylor formula and the -Rodrigues' formula (the formula used to construct -orthogonal polynomials).

Generalizations

Post Quantum Calculus

Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator:

Hahn difference

Wolfgang Hahn introduced the following operator (Hahn difference):

When this operator reduces to -derivative, and when it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems.

β-derivative

-derivative is an operator defined as follows:

In the definition, is a given interval, and is any continuous function that strictly monotonically increases (i.e. ). When then this operator is -derivative, and when this operator is Hahn difference.