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 Exton (1983).

Definition

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

$\int f(x)\,{\rm {d}}_{q}x=(1-q)\,x\sum _{k=0}^{\infty }q^{k}f(q^{k}x).$

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

$\int f(x)\,D_{q}g\,{\rm {d}}_{q}x=(1-q)\,x\sum _{k=0}^{\infty }q^{k}f(q^{k}x)\,D_{q}g(q^{k}x)=(1-q)\,x\sum _{k=0}^{\infty }q^{k}f(q^{k}x){\tfrac {g(q^{k}x)-g(q^{k+1}x)}{(1-q)q^{k}x}},$ 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.

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.
• Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538