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:
Consistent with this is the definition for
[math]\displaystyle{ \int_0^\infty f(x)\,{\rm d}_q x = (1-q)\sum_{k=-\infty}^{\infty}q^k f(q^k ). }[/math]
More generally, if g(x) is another function and Dqg denotes its q-derivative, we can formally write
or
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
Notes
- ↑ Exton, H (1979). "Basic Fourier series". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 369 (1736): 115–136. doi:10.1098/rspa.1979.0155. Bibcode: 1979RSPSA.369..115E.
- ↑ Kempf, A; Majid, Shahn (1994). "Algebraic q-Integration and Fourier Theory on Quantum and Braided Spaces". Journal of Mathematical Physics 35 (12): 6802–6837. doi:10.1063/1.530644. Bibcode: 1994JMP....35.6802K.
- ↑ 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 xn", 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.
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