# 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:

More generally, if *g*(*x*) is another function and *D*_{q}*g* denotes its *q*-derivative, we can formally write

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 If is bounded on the interval for some then the Jackson integral converges to a function on which is a *q*-antiderivative of Moreover, is continuous at with and is a unique antiderivative of in this class of functions.^{[1]}

## Notes

- ↑ 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 x
_{n}",*Proc. R. Soc.***74**64–72. - Jackson F H (1910), "On q-definite integrals",
*Q. J. Pure Appl. Math.***41**193–203.