# 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:

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

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

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

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.

### Theorem

Suppose that ${\displaystyle 0 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.[1]

## Notes

1. Kac-Cheung, Theorem 19.1.

## References

• Victor Kac, Pokman Cheung, Quantum Calculus, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8
• Jackson F H (1904), "A generalization of the functions Γ(n) and xn", Proc. R. Soc. 74 64–72.
• Jackson F H (1910), "On q-definite integrals", Q. J. Pure Appl. Math. 41 193–203.