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.

Definition

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

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.

Theorem

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.[1]

Notes

  1. 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.

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