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 . The q-derivative of a function f(x) 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 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). 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: 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 (the formula used to construct -orthogonal polynomials). Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator: Wolfgang Hahn introduced the following operator (Hahn difference): 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.