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 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:


 * $$ \int_0^a f(x)\,{\rm d}_q x = (1-q)\,a\sum_{k=0}^{\infty}q^k f(q^k a). $$

Consistent with this is the definition for $$ a \to \infty $$

$$ \int_0^\infty f(x)\,{\rm d}_q x = (1-q)\sum_{k=-\infty}^{\infty}q^k f(q^k ). $$

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


 * $$ \int f(x)\,D_q g\,{\rm d}_q x = (1-q)\,x\sum_{k=0}^{\infty}q^k f(q^k x)\,D_q g(q^k x) = (1-q)\,x\sum_{k=0}^{\infty}q^k f(q^k x)\tfrac{g(q^{k}x)-g(q^{k+1}x)}{(1-q)q^k x}, $$ or


 * $$ \int f(x)\,{\rm d}_q g(x) = \sum_{k=0}^{\infty} f(q^k x)\cdot(g(q^{k}x)-g(q^{k+1}x)), $$

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 (see ).

Theorem
Suppose that $$0<q<1.$$ If $$|f(x)x^\alpha|$$ is bounded on the interval $$[0,A)$$ for some $$0\leq\alpha<1, $$ 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.