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.

Definition

Let f be a function of a real variable x. The Jackson integral of f is defined by the following series expansion:
More generally, if g 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 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.