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.

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_qg denotes its q-derivative, we can formally write

giving a q-analogue of the Riemann–Stieltjes integral.

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,

Suppose that ${\displaystyle 0<q<1.}$ 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.

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