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


 * $$\left(\frac{d}{dx}\right)_q f(x)=\frac{f(qx)-f(x)}{qx-x}.$$

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
 * $$D_q= \frac{1}{x} ~ \frac{q^{d \over d (\ln x)} -1}{q-1} ~, $$

which goes to the plain derivative, $$D_q \to \frac{d}{dx}$$ 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,


 * $$\displaystyle D_q (f(x)/g(x)) = \frac{g(x)D_q f(x) - f(x)D_q g(x)}{g(qx)g(x)},\quad g(x)g(qx)\neq 0. $$

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


 * $$\displaystyle D_q f(g(x)) = D_{q^k}(f)(g(x))D_q(g)(x).$$

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:


 * $$\left(\frac{d}{dz}\right)_q z^n = \frac{1-q^n}{1-q} z^{n-1} =

[n]_q z^{n-1}$$

where $$[n]_q$$ is the q-bracket of n. Note that $$\lim_{q\to 1}[n]_q = n$$ 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{f^{(n)}(0)}{n!} \frac{(q;q)_n}{(1-q)^n}= \frac{f^{(n)}(0)}{n!} [n]!_q $$

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


 * $$\displaystyle D_q(f(x)) = \sum_{k=0}^{\infty}\frac{(q-1)^k}{(k+1)!} x^k f^{(k+1)}(x).$$

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


 * $$f(z)=\sum_{n=0}^\infty f^{(n)}(0)\,\frac{z^n}{n!} = \sum_{n=0}^\infty (D^n_q f)(0)\,\frac{z^n}{[n]!_q}.$$

Higher order q-derivatives
The following representation for higher order $$q$$-derivatives is known:
 * $$D_q^nf(x)=\frac{1}{(1-q)^nx^n}\sum_{k=0}^n(-1)^k\binom{n}{k}_q q^{\binom{k}{2}-(n-1)k}f(q^kx).$$

$$\binom{n}{k}_q$$ is the $$q$$-binomial coefficient. By changing the order of summation as $$r=n-k$$, we obtain the next formula:
 * $$D_q^nf(x)=\frac{(-1)^n q^{-\binom{n}{2}}}{(1-q)^nx^n}\sum_{r=0}^n(-1)^r\binom{n}{r}_q q^{\binom{r}{2}}f(q^{n-r}x).$$

Higher order $$q$$-derivatives are used to $$q$$-Taylor formula and the $$q$$-Rodrigues' formula (the formula used to construct $$q$$-orthogonal polynomials).

Post Quantum Calculus
Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator:
 * $$D_{p,q}f(x):=\frac{f(px)-f(qx)}{(p-q)x},\quad x\neq 0.$$

Hahn difference
Wolfgang Hahn introduced the following operator (Hahn difference):
 * $$D_{q,\omega}f(x):=\frac{f(qx+\omega)-f(x)}{(q-1)x+\omega},\quad 00.$$

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:
 * $$D_\beta f(t):=\frac{f(\beta(t))-f(t)}{\beta(t)-t},\quad\beta\neq t,\quad\beta:I\to I.$$

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.