q-derivative

Short description: Q-analog of the ordinary 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 (Chung Chung).

Definition

The q-derivative of a function f(x) is defined as

[math]\displaystyle{ \left(\frac{d}{dx}\right)_q f(x)=\frac{f(qx)-f(x)}{qx-x}. }[/math]

It is also often written as [math]\displaystyle{ D_qf(x) }[/math]. 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

[math]\displaystyle{ D_q= \frac{1}{x} ~ \frac{q^{d~~~ \over d (\ln x)} -1}{q-1} ~, }[/math]

which goes to the plain derivative, [math]\displaystyle{ D_q \to \frac{d}{dx} }[/math] as [math]\displaystyle{ q \to 1 }[/math].

It is manifestly linear,

[math]\displaystyle{ \displaystyle D_q (f(x)+g(x)) = D_q f(x) + D_q g(x)~. }[/math]

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

[math]\displaystyle{ \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). }[/math]

Similarly, it satisfies a quotient rule,

[math]\displaystyle{ \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. }[/math]

There is also a rule similar to the chain rule for ordinary derivatives. Let [math]\displaystyle{ g(x) = c x^k }[/math]. Then

[math]\displaystyle{ \displaystyle D_q f(g(x)) = D_{q^k}(f)(g(x))D_q(g)(x). }[/math]

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:

[math]\displaystyle{ \left(\frac{d}{dz}\right)_q z^n = \frac{1-q^n}{1-q} z^{n-1} = [n]_q z^{n-1} }[/math]

where [math]\displaystyle{ [n]_q }[/math] is the q-bracket of n. Note that [math]\displaystyle{ \lim_{q\to 1}[n]_q = n }[/math] so the ordinary derivative is regained in this limit.

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

[math]\displaystyle{ (D^n_q f)(0)= \frac{f^{(n)}(0)}{n!} \frac{(q;q)_n}{(1-q)^n}= \frac{f^{(n)}(0)}{n!} [n]!_q }[/math]

provided that the ordinary n-th derivative of f exists at x = 0. Here, [math]\displaystyle{ (q;q)_n }[/math] is the q-Pochhammer symbol, and [math]\displaystyle{ [n]!_q }[/math] is the q-factorial. If [math]\displaystyle{ f(x) }[/math] is analytic we can apply the Taylor formula to the definition of [math]\displaystyle{ D_q(f(x)) }[/math] to get

[math]\displaystyle{ \displaystyle D_q(f(x)) = \sum_{k=0}^{\infty}\frac{(q-1)^k}{(k+1)!} x^k f^{(k+1)}(x). }[/math]

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

[math]\displaystyle{ 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}. }[/math]

Higher order q-derivatives

The following representation for higher order [math]\displaystyle{ q }[/math]-derivatives is known:

[math]\displaystyle{ 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). }[/math]

[math]\displaystyle{ \binom{n}{k}_q }[/math] is the [math]\displaystyle{ q }[/math]-binomial coefficient. By changing the order of summation as [math]\displaystyle{ r=n-k }[/math], we obtain the next formula:

[math]\displaystyle{ 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). }[/math]

Higher order [math]\displaystyle{ q }[/math]-derivatives are used to [math]\displaystyle{ q }[/math]-Taylor formula and the [math]\displaystyle{ q }[/math]-Rodrigues' formula (the formula used to construct [math]\displaystyle{ q }[/math]-orthogonal polynomials).

Generalizations

Post Quantum Calculus

Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator:

[math]\displaystyle{ D_{p,q}f(x):=\frac{f(px)-f(qx)}{(p-q)x},\quad x\neq 0. }[/math]

Hahn difference

Wolfgang Hahn introduced the following operator (Hahn difference):

[math]\displaystyle{ D_{q,\omega}f(x):=\frac{f(qx+\omega)-f(x)}{(q-1)x+\omega},\quad 0\lt q\lt 1,\quad\omega\gt 0. }[/math]

When [math]\displaystyle{ \omega\to0 }[/math] this operator reduces to [math]\displaystyle{ q }[/math]-derivative, and when [math]\displaystyle{ q\to1 }[/math] it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems.

β-derivative

[math]\displaystyle{ \beta }[/math]-derivative is an operator defined as follows:

[math]\displaystyle{ D_\beta f(t):=\frac{f(\beta(t))-f(t)}{\beta(t)-t},\quad\beta\neq t,\quad\beta:I\to I. }[/math]

In the definition, [math]\displaystyle{ I }[/math] is a given interval, and [math]\displaystyle{ \beta(t) }[/math] is any continuous function that strictly monotonically increases (i.e. [math]\displaystyle{ t\gt s\rightarrow\beta(t)\gt \beta(s) }[/math]). When [math]\displaystyle{ \beta(t)=qt }[/math] then this operator is [math]\displaystyle{ q }[/math]-derivative, and when [math]\displaystyle{ \beta(t)=qt+\omega }[/math] 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

Bibliography

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