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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.

Contents

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:

∫ f ( x ) d q x = ( 1 − q ) x ∑ k = 0 ∞ q k f ( q k x ) .

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

∫ f ( x ) D q g d q x = ( 1 − q ) x ∑ k = 0 ∞ q k f ( q k x ) D q g ( q k x ) = ( 1 − q ) x ∑ k = 0 ∞ q k f ( q k x ) g ( q k x ) − g ( q k + 1 x ) ( 1 − q ) q k x , or ∫ f ( x ) d q g ( x ) = ∑ 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,

Theorem

Suppose that 0 < q < 1. If | f ( x ) x α | is bounded on the interval [ 0 , A ) for some 0 ≤ α < 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.

References

Jackson integral Wikipedia
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