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 ^[1] 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:

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

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

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

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

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

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

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 ^[2]).

Theorem

Suppose that ${\displaystyle 0<q<1.}$ If ${\displaystyle |f(x)x^{\alpha }|}$ is bounded on the interval ${\displaystyle [0,A)}$ for some ${\displaystyle 0\leq \alpha <1,}$ then the Jackson integral converges to a function ${\displaystyle F(x)}$ on ${\displaystyle [0,A)}$ which is a q-antiderivative of ${\displaystyle f(x).}$ Moreover, ${\displaystyle F(x)}$ is continuous at ${\displaystyle x=0}$ with ${\displaystyle F(0)=0}$ and is a unique antiderivative of ${\displaystyle f(x)}$ in this class of functions.^[3]

Notes

1. Exton, H (1979). "Basic Fourier series". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 369 (1736): 115–136. Bibcode:1979RSPSA.369..115E. doi:10.1098/rspa.1979.0155. S2CID 120587254.
2. Kempf, A; Majid, Shahn (1994). "Algebraic q-Integration and Fourier Theory on Quantum and Braided Spaces". Journal of Mathematical Physics. 35 (12): 6802–6837. arXiv:hep-th/9402037. Bibcode:1994JMP....35.6802K. doi:10.1063/1.530644. S2CID 16930694.
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.
• Exton, Harold (1983). Q-hypergeometric functions and applications. Chichester [West Sussex]: E. Horwood. ISBN 978-0470274538.