Jackson integral

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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.


Let f(x) be a function of a real variable x. The Jackson integral of f is defined by the following series expansion:

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


giving a q-analogue of the Riemann–Stieltjes integral.

Jackson integral as q-antiderivative[edit]

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,[1] [2]


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]


  1. ^ Kempf, A., Majid, S. (1994). "Algebraic q‐integration and Fourier theory on quantum and braided spaces". J. Math. Phys. (35): 6802. doi:10.1063/1.530644. Retrieved 24 April 2015. 
  2. ^ Kempf, A., Majid, S. "Algebraic q‐integration and Fourier theory on quantum and braided spaces, arxiv version" (PDF). Arxiv. Retrieved 24 April 2015. 
  3. ^ Kac-Cheung, Theorem 19.1.


  • 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.