Jackson integral

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In mathematics, the Jackson integral is a series in the theory of special functions that expresses the operation inverse to q-differentiation.

1 Definition
2 Jackson integral as q-antiderivative
- 2.1 Theorem
3 Notes
4 References

[edit] Definition

Let f(x) be a function of a real variable x. The Jackson integral of f is defined by the following series expansion:

$\int f(x) d_q x = (1-q)x\sum_{k=0}^{\infty}q^k f(q^k x).$

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

$\int f(x) D_q g d_q x = (1-q)x\sum_{k=0}^{\infty}q^k f(q^k x) D_q g(q^k x) = (1-q)x\sum_{k=0}^{\infty}q^k f(q^k x)\frac{g(q^{k}x)-g(q^{k+1}x)}{(1-q)q^k x},$ or

$\int f(x) d_q g(x) = \sum_{k=0}^{\infty} f(q^k x)(g(q^{k}x)-g(q^{k+1}x)),$

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

[edit] 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.

[edit] Theorem

Suppose that $0 < q < 1.$ If $| f (x) x α |$ 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.^[1]