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

:$\left(\frac\right)_q f(x)=\frac.$

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
:$D_q= \frac ~ \frac ~,$
which goes to the plain derivative, $D_q \to \frac$ as $q \to 1$.

It is manifestly linear,
:$\displaystyle D_q (f(x)+g(x)) = D_q f(x) + D_q g(x)~.$

It has a product rule analogous to the ordinary derivative product rule, with two equivalent forms

:$\displaystyle D_q (f(x)g(x)) = g(x)D_q f(x) + f(qx)D_q g(x) = g(qx)D_q f(x) + f(x)D_q g(x).$

Similarly, it satisfies a quotient rule,

:$\displaystyle D_q (f(x)/g(x)) = \frac,\quad g(x)g(qx)\neq 0.$

There is also a rule similar to the chain rule for ordinary derivatives. Let $g(x) = c x^k$. Then

:$\displaystyle D_q f(g(x)) = D_(f)(g(x))D_q(g)(x).$

The eigenfunction of the ''q''-derivative is the ''q''-exponential ''e_q''(''x'').

Relationship to ordinary derivatives

''Q''-differentiation resembles ordinary differentiation, with curious differences. For example, the ''q''-derivative of the monomial is:

:\left(\frac\right)_q z^n = \frac z^ =
q z^

where q is the ''q''-bracket of ''n''. Note that \lim_ q = n so the ordinary derivative is regained in this limit.

The ''n''-th ''q''-derivative of a function may be given as:

:(D^n_q f)(0)=
\frac \frac=
\frac _q

provided that the ordinary ''n''-th derivative of ''f'' exists at ''x'' = 0. Here, $(q;q)_n$ is the ''q''-Pochhammer symbol, and _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

:$\displaystyle D_q(f(x)) = \sum_^\frac x^k f^(x).$

A ''q''-analog of the Taylor expansion of a function about zero follows:

:$f(z)=\sum_^\infty f^(0)\,\frac = \sum_^\infty (D^n_q f)(0)\,\frac.$

Higher order ''q''-derivatives

The following representation for higher order $q$-derivatives is known:
:$D_q^nf(x)=\frac\sum_^n(-1)^k\binom_q q^f(q^kx).$
$\binom_q$ is the $q$-binomial coefficient. By changing the order of summation as $r=n-k$, we obtain the next formula:
:$D_q^nf(x)=\frac\sum_^n(-1)^r\binom_q q^f(q^x).$

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:
:$D_f(x):=\frac,\quad x\neq 0.$

Hahn difference

Wolfgang Hahn introduced the following operator (Hahn difference):
:D_f(x):=\frac,\quad 00.

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:Auch, T. (2013): ''Development and Application of Difference and Fractional Calculus on Discrete Time Scales''. PhD thesis, University of Nebraska-Lincoln.
:$D_\beta f(t):=\frac,\quad\beta\neq t,\quad\beta:I\to I.$

In the definition, $I$ is a given interval, and $\beta(t)$ is any continuous function that strictly monotonically increases (i.e. $t>s\rightarrow\beta(t)>\beta(s)$). When $\beta(t)=qt$ then this operator is $q$-derivative, and when $\beta(t)=qt+\omega$ 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

Citations

Bibliography

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{{refend

Differential calculus
Generalizations of the derivative
Linear operators in calculus
Q-analogs