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
It is also often written as formula_2. 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 formula_4 as formula_5.
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 formula_9. 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 formula_12 is the "q"-bracket of "n". Note that formula_13 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, formula_15 is the "q"-Pochhammer symbol, and formula_16 is the "q"-factorial. If formula_17 is analytic we can apply the Taylor formula to the definition of formula_18 to get
A "q"-analog of the Taylor expansion of a function about zero follows:
Higher order formula_21-derivatives.
Th following representation for higher order formula_21-derivatives is known:
formula_24 is the formula_21-binomial coefficient. By changing the order of summation as formula_26, we obtain the next formula:
Higher order formula_21-derivatives are used to formula_21-Taylor formula and the formula_21-Rodrigues' formula (the formula used to construct formula_21-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 formula_34 this operator reduces to formula_21-derivative, and when formula_36 it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems.
formula_37-derivative.
formula_37-derivative is an operator defined as follows:
In the definition, formula_40 is a given interval, and formula_41 is any continuous function that strictly monotonically increases (i.e formula_42). When formula_43 then this operator is formula_21-derivative, and when formula_45 this operator is Hahn difference.