Quantum calculus

Builds on q-derivative Jackson integral Function

Definition

Quantum calculus is the study of functions through the following two difference operators in place of the derivative.

For a fixed $q \in \C \setminus \set{0, 1}$ , the $q$ -derivative of $f$ is $D_q f(x) = \dfrac{f(qx) - f(x)}{(q - 1)x}$ for $x \ne 0$ ; this is the q-derivative.
For a fixed $h \in \C \setminus \set{0}$ , the $h$ -derivative of $f$ is $D_h f(x) = \dfrac{f(x + h) - f(x)}{h}$ .

The corresponding antiderivatives are the Jackson integral $\int_0^x f(t) \, d_q t = (1 - q) x \sum_{k \ge 0} q^k f(q^k x)$ , for $0 < \abs{q} < 1$ when the series converges, and the sum $h \sum_{k=0}^{n-1} f(a + kh)$ over a grid of step $h$ . The part built on $D_q$ is called $q$ -calculus, and the part built on $D_h$ is called $h$ -calculus.

Remarks

Example: for $f(x) = x^n$ with $n \ge 1$ , $D_q f(x) = \dfrac{q^n x^n - x^n}{(q - 1)x} = [n]_q x^{n-1}$ , where $[n]_q = 1 + q + \cdots + q^{n-1}$ .

Non-example: the ordinary derivative $f'(x) = \lim_{h \to 0} (f(x + h) - f(x))/h$ is not an operator of quantum calculus, because it involves a limit rather than a fixed step $q$ or $h$ . It is the common limit of $D_q$ as $q \to 1$ and of $D_h$ as $h \to 0$ for differentiable $f$ .

The $q$ -derivative satisfies the product rule $D_q(fg)(x) = f(qx) D_q g(x) + g(x) D_q f(x)$ . The Jackson integral inverts $D_q$ : if $F(x) = \int_0^x f(t) \, d_q t$ converges at $x$ , then $D_q F(x) = f(x)$ .

Reading path

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Function concept
Jackson integral concept
q-derivative concept
Quantum calculus definition

Tag	`TTR52N9E7M`
URI	`m2://TTR52N9E7M`
Kind	definition
Writing	W0 stub
MSC	05A30, 39A13
Defines	Quantum calculus
Uses	q-derivative, Jackson integral, Function

Quantum calculus

Definition

Remarks

Reading path

Further reading

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