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 It is also often written as D_qf(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, as q \to 1. 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 eq(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 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:

Higher order q-derivatives

The following representation for higher order q-derivatives is known: is the q-binomial coefficient. By changing the order of summation as r=n-k, we obtain the next formula: Higher order q-derivatives are used to q-Taylor formula and the q-Rodrigues' formula (the formula used to construct q-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 \omega\to0 this operator reduces to q-derivative, and when q\to1 it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems.

β-derivative

\beta-derivative is an operator defined as follows: In the definition, I is a given interval, and \beta(t) is any continuous function that strictly monotonically increases (i.e. ). When \beta(t)=qt then this operator is q-derivative, and when this operator is Hahn difference.

Applications

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

Citations