# 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.

## Definition

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

$\int f(x)d_{q}x=(1-q)x\sum _{k=0}^{\infty }q^{k}f(q^{k}x).$ More generally, if g(x) is another function and Dqg denotes its q-derivative, we can formally write

$\int f(x)D_{q}gd_{q}x=(1-q)x\sum _{k=0}^{\infty }q^{k}f(q^{k}x)D_{q}g(q^{k}x)=(1-q)x\sum _{k=0}^{\infty }q^{k}f(q^{k}x){\frac {g(q^{k}x)-g(q^{k+1}x)}{(1-q)q^{k}x}},$ or
$\int f(x)d_{q}g(x)=\sum _{k=0}^{\infty }f(q^{k}x)(g(q^{k}x)-g(q^{k+1}x)),$ 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.