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

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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  

 

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

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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

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Suppose that   If   is bounded on the interval   for some   then the Jackson integral converges to a function   on   which is a q-antiderivative of   Moreover,   is continuous at   with   and is a unique antiderivative of   in this class of functions.[3]

Notes

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  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

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  • 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.