Jackson integral explained

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 .

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:

\int ^{a} _{0} f(x){\rm d} _{q} x = ( 1- q)a\sum ^{inf ty} _{k =0} q ^{k} f(q ^{k} a).

Consistent with this is the definition for

a \to inf ty \int ^{inf ty} _{0} f(x){\rm d} _{q} x = ( 1- q)\sum ^{inf ty} _{k =-inf ty} 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{\rm d} _{q} x = ( 1- q)x\sum ^{inf ty} _{k =0} q ^{k} f(q ^{k} x)D _{q} g(q ^{k} x) = ( 1- q)x\sum ^{inf ty} _{k =0} 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 ^{inf ty} _{k =0} 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 ^[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. Exton. H. Basic Fourier series. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 369. 1736. 115–136. 1979. 10.1098/rspa.1979.0155. 1979RSPSA.369..115E. 120587254.
2. Kempf. A. Algebraic q-Integration and Fourier Theory on Quantum and Braided Spaces. Journal of Mathematical Physics. 35. 12. 6802–6837. Majid. Shahn. 1994. hep-th/9402037. 1994JMP....35.6802K. 10.1063/1.530644. 16930694.
3. Kac-Cheung, Theorem 19.1.

References

• Victor Kac, Pokman Cheung, Quantum Calculus, Universitext, Springer-Verlag, 2002.
• 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.
• Book: Exton . Harold . Q-hypergeometric functions and applications . 1983 . E. Horwood . Chichester [West Sussex] . 978-0470274538.