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.

Definition

The q-derivative of a function f 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 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.

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

Th 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.

Generalizations

Post Quantum Calculus

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

Hahn difference

introduced the following operator :
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.

\beta-derivative

-derivative is an operator defined as follows:
In the definition, is a given interval, and is any continuous function that strictly monotonically increases. When then this operator is -derivative, and when this operator is Hahn difference.