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 Exton (1983).

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:

$\underset{0}{\overset{a}{\int }}f(x){\mathrm{d}}_{q}x=(1-q)a\sum _{k=0}^{\mathrm{\infty }}{q}^{k}f(q^{k}a).$

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

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

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

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

$\int f(x){\mathrm{d}}_{q}g(x)=\sum _{k=0}^{\mathrm{\infty }}f(q^{k}x)\cdot (g(q^{k}x)-g(q^{k+1}x)),$

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 $0\le \alpha <1,$ 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, Harold (1983). Q-hypergeometric functions and applications. Chichester [West Sussex]: E. Horwood. ISBN 978-0470274538.