Jackson integral

In q-analog theory, the Jackson integral series in the theory of special functions that expresses the operation inverse to q-differentiation.

The Jackson integral was introduced by Frank Hilton Jackson. For methods of numerical evaluation, see ^[1] and Exton (1983).

Definition

Let f(x) be a function of a real variable x. For a a real variable, the Jackson integral of f is defined by the following series expansion:

${\displaystyle \int _{0}^{a}f(x)\,{\rm {d}}_{q}x=(1-q)\,a\sum _{k=0}^{\infty }q^{k}f(q^{k}a).}$

Consistent with this is the definition for ${\displaystyle a\to \infty }$

${\displaystyle \int _{0}^{\infty }f(x)\,{\rm {d}}_{q}x=(1-q)\sum _{k=-\infty }^{\infty }q^{k}f(q^{k}).}$

More generally, if g(x) is another function and D_qg denotes its q-derivative, we can formally write

${\displaystyle \int f(x)\,D_{q}g\,{\rm {d}}_{q}x=(1-q)\,x\sum _{k=0}^{\infty }q^{k}f(q^{k}x)\,D_{q}g(q^{k}x)=(1-q)\,x\sum _{k=0}^{\infty }q^{k}f(q^{k}x){\tfrac {g(q^{k}x)-g(q^{k+1}x)}{(1-q)q^{k}x}},}$ or

${\displaystyle \int f(x)\,{\rm {d}}_{q}g(x)=\sum _{k=0}^{\infty }f(q^{k}x)\cdot (g(q^{k}x)-g(q^{k+1}x)),}$

giving a q-analogue of the Riemann–Stieltjes integral.

Jackson integral as q-antiderivative

Just as the ordinary antiderivative of a continuous function can be represented by its Riemann integral, it is possible to show that the Jackson integral gives a unique q-antiderivative within a certain class of functions (see ^[2]).

Theorem

Suppose that ${\displaystyle 0<q<1.}$ If ${\displaystyle |f(x)x^{\alpha }|}$ is bounded on the interval ${\displaystyle [0,A)}$ for some ${\displaystyle 0\leq \alpha <1,}$ then the Jackson integral converges to a function ${\displaystyle F(x)}$ on ${\displaystyle [0,A)}$ which is a q-antiderivative of ${\displaystyle f(x).}$ Moreover, ${\displaystyle F(x)}$ is continuous at ${\displaystyle x=0}$ with ${\displaystyle F(0)=0}$ and is a unique antiderivative of ${\displaystyle f(x)}$ in this class of functions.^[3]