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 and.

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: Consistent with this is the definition for More generally, if g(x) is another function and Dqg 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 (see ).

Theorem

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