Q-derivative on arvau

Wikipedia / 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(x) is defined 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

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,

. 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:

is the q-bracket of 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 is the q-Pochhammer symbol is the q-factorial is analytic we can apply the Taylor formula to get

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

Higher order q-derivatives

-derivatives is known:

, we obtain the next formula:

-Rodrigues' formula-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):

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:

is any continuous function this operator is Hahn difference.

Applications

The q-calculus has been used in machine learning for designing stochastic activation functions.