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 Chung et al. (1994).

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 $D_{q}f(x)$. 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

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

which goes to the plain derivative, $D_q\to \frac{d}{dx}$ as $q\to 1$.

It is manifestly linear,

$D_q(f(x)+g(x))=D_{q}f(x)+D_{q}g(x).$

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

$D_q(f(x)g(x))=g(x)D_{q}f(x)+f(qx)D_{q}g(x)=g(qx)D_{q}f(x)+f(x)D_{q}g(x).$

Similarly, it satisfies a quotient rule,

$D_q(f(x)/g(x))=\frac{g(x)D_{q}f(x)-f(x)D_{q}g(x)}{g(qx)g(x)},g(x)g(qx)\ne 0.$

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

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

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

Relationship to ordinary derivatives

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

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

where $[n{]}_q$ is the q-bracket of n. Note that $\underset{q\to 1}{\lim }[n{]}_q=n$ so the ordinary derivative is regained in this limit.

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

$(D_q^{n}f)(0)=\frac{f^{(n)}(0)}{n!}\frac{(q;q{)}_n}{(1-q{)}^n}=\frac{f^{(n)}(0)}{n!}[n]{!}_q$

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

$D_q(f(x))=\sum _{k=0}^{\mathrm{\infty }}\frac{(q-1{)}^k}{(k+1)!}{x}^k{f}^{(k+1)}(x).$

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

$f(z)=\sum _{n=0}^{\mathrm{\infty }}{f}^{(n)}(0)\frac{z^n}{n!}=\sum _{n=0}^{\mathrm{\infty }}(D_q^{n}f)(0)\frac{z^n}{[n]{!}_q}.$

Higher order q-derivatives

The following representation for higher order $q$-derivatives is known:

$D_q^{n}f(x)=\frac{1}{(1-q{)}^n{x}^n}\sum _{k=0}^n(-1{)}^k{(\genfrac{}{}{0ex}{}{n}{k})}_q{q}^{(\genfrac{}{}{0ex}{}{k}{2})-(n-1)k}f(q^{k}x).$

${(\genfrac{}{}{0ex}{}{n}{k})}_q$ is the $q$-binomial coefficient. By changing the order of summation as $r=n-k$, we obtain the next formula:

$D_q^{n}f(x)=\frac{(-1{)}^n{q}^{-(\genfrac{}{}{0ex}{}{n}{2})}}{(1-q{)}^n{x}^n}\sum _{r=0}^n(-1{)}^r{(\genfrac{}{}{0ex}{}{n}{r})}_q{q}^{(\genfrac{}{}{0ex}{}{r}{2})}f(q^{n-r}x).$

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

Generalizations

Post Quantum Calculus

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

$D_{p,q}f(x):=\frac{f(px)-f(qx)}{(p-q)x},x\ne 0.$

Hahn difference

Wolfgang Hahn introduced the following operator (Hahn difference):

$D_{q,\omega }f(x):=\frac{f(qx+\omega )-f(x)}{(q-1)x+\omega },0<q<1,\omega >0.$

When $\omega \to 0$ this operator reduces to $q$-derivative, and when $q\to 1$ 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:

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

In the definition, $I$ is a given interval, and $\beta (t)$ is any continuous function that strictly monotonically increases (i.e. $t>s\to \beta (t)>\beta (s)$). 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