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).
The q-derivative of a function f(x) is defined as[1][2][3] 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,
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.
Then The eigenfunction of the q-derivative is the q-exponential eq(x).
Q-differentiation resembles ordinary differentiation, with curious differences.
For example, the q-derivative of the monomial is:[2] where
so the ordinary derivative is regained in this limit.
The n-th q-derivative of a function may be given as:[3] provided that the ordinary n-th derivative of f exists at x = 0.
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:[2] The following representation for higher order
By changing the order of summation as
, we obtain the next formula:[4][6] Higher order
Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator:[7][8] Wolfgang Hahn introduced the following operator (Hahn difference):[9][10] When
this operator reduces to
it reduces to forward difference.
This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems.
-derivative is an operator defined as follows:[14][15] In the definition,
is any continuous function that strictly monotonically increases (i.e.
β ( t ) = q t + ω
this operator is Hahn difference.
The q-calculus has been used in machine learning for designing stochastic activation functions.