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.

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.

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,^{[1] [2]}

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.^[3]

Notes

1. Kempf, A., Majid, S. (1994). "Algebraic q‐integration and Fourier theory on quantum and braided spaces". J. Math. Phys. (35): 6802. doi:10.1063/1.530644. Retrieved 24 April 2015.
2. Kempf, A., Majid, S. "Algebraic q‐integration and Fourier theory on quantum and braided spaces, arxiv version" (PDF). Arxiv. Retrieved 24 April 2015.
3. Kac-Cheung, Theorem 19.1.

References

• Victor Kac, Pokman Cheung, Quantum Calculus, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8
• Jackson F H (1904), "A generalization of the functions Γ(n) and x_n", Proc. R. Soc. 74 64–72.
• Jackson F H (1910), "On q-definite integrals", Q. J. Pure Appl. Math. 41 193–203.