Jackson integral

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In q-analog theory, the Jackson integral or q-integral is a 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:

∫ 0 a f ( x ) d q x = ( 1 − q ) a ∑ k = 0 ∞ q k f ( q k a ) . {\displaystyle \int _{0}^{a}f(x)\,{\rm {d}}_{q}x=(1-q)\,a\sum _{k=0}^{\infty }q^{k}f(q^{k}a).} {\displaystyle \int _{0}^{a}f(x)\,{\rm {d}}_{q}x=(1-q)\,a\sum _{k=0}^{\infty }q^{k}f(q^{k}a).}

Consistent with this is the definition for a → ∞ {\displaystyle a\to \infty } {\displaystyle a\to \infty }

∫ 0 ∞ f ( x ) d q x = ( 1 − q ) ∑ k = − ∞ ∞ q k f ( q k ) . {\displaystyle \int _{0}^{\infty }f(x)\,{\rm {d}}_{q}x=(1-q)\sum _{k=-\infty }^{\infty }q^{k}f(q^{k}).} {\displaystyle \int _{0}^{\infty }f(x)\,{\rm {d}}_{q}x=(1-q)\sum _{k=-\infty }^{\infty }q^{k}f(q^{k}).}

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 , {\displaystyle \int f(x)\,D_{q}g\,{\rm {d}}_{q}x=(1-q)\,x\sum _{k=0}^{\infty }q^{k}f(q^{k}x)\,D_{q}g(q^{k}x)=(1-q)\,x\sum _{k=0}^{\infty }q^{k}f(q^{k}x){\tfrac {g(q^{k}x)-g(q^{k+1}x)}{(1-q)q^{k}x}},} {\displaystyle \int f(x)\,D_{q}g\,{\rm {d}}_{q}x=(1-q)\,x\sum _{k=0}^{\infty }q^{k}f(q^{k}x)\,D_{q}g(q^{k}x)=(1-q)\,x\sum _{k=0}^{\infty }q^{k}f(q^{k}x){\tfrac {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 ) ) , {\displaystyle \int f(x)\,{\rm {d}}_{q}g(x)=\sum _{k=0}^{\infty }f(q^{k}x)\cdot (g(q^{k}x)-g(q^{k+1}x)),} {\displaystyle \int f(x)\,{\rm {d}}_{q}g(x)=\sum _{k=0}^{\infty }f(q^{k}x)\cdot (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 [2]).

Theorem

Suppose that 0 < q < 1. {\displaystyle 0<q<1.} {\displaystyle 0<q<1.} If | f ( x ) x α | {\displaystyle |f(x)x^{\alpha }|} {\displaystyle |f(x)x^{\alpha }|} is bounded on the interval [ 0 , A ) {\displaystyle [0,A)} {\displaystyle [0,A)} for some 0 ≤ α < 1 , {\displaystyle 0\leq \alpha <1,} {\displaystyle 0\leq \alpha <1,} then the Jackson integral converges to a function F ( x ) {\displaystyle F(x)} {\displaystyle F(x)} on [ 0 , A ) {\displaystyle [0,A)} {\displaystyle [0,A)} which is a q-antiderivative of f ( x ) . {\displaystyle f(x).} {\displaystyle f(x).} Moreover, F ( x ) {\displaystyle F(x)} {\displaystyle F(x)} is continuous at x = 0 {\displaystyle x=0} {\displaystyle x=0} with F ( 0 ) = 0 {\displaystyle F(0)=0} {\displaystyle F(0)=0} and is a unique antiderivative of f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} in this class of functions.[3]

Notes

  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

  • 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.