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 . 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).
Relationship to ordinary derivatives 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:
Higher order q-derivatives The 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).
Generalizations Post Quantum Calculus Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator:Gupta V., Rassias T.M., Agrawal P.N., Acu A.M. (2018) Basics of Post-Quantum Calculus. In: Recent Advances in Constructive Approximation Theory. SpringerOptimization and Its Applications, vol 138. Springer.
Hahn difference Wolfgang Hahn introduced the following operator (Hahn difference):Hahn, W. (1949). Math. Nachr. 2: 4-34.Hahn, W. (1983) Monatshefte Math. 95: 19-24.
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.Kwon, K.; Lee, D.; Park, S.; Yoo, B.: Kyungpook Math. J. 38, 259-281 (1998).Alvarez-Nodarse, R.: J. Comput. Appl. Math. 196, 320-337 (2006).
β-derivative -derivative is an operator defined as follows:Auch, T. (2013): Development and Application of Difference and Fractional Calculus on Discrete Time Scales. PhD thesis, University of Nebraska-Lincoln.
In the definition, is a given interval, and is any continuous function that strictly monotonically increases (i.e. ). When then this operator is -derivative, and when this operator is Hahn difference.
Applications The q-calculus has been used in machine learning for designing stochastic activation functions.
See also Derivative (generalizations) Jackson integral Q-exponential Q-difference polynomials Quantum calculus Tsallis entropy
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