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:

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

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

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

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

${\displaystyle \int f(x)\,D_{q}g\,{\rm {d}}_{q}x=(1-q)\,x\sum _{k=0}^{\infty }q^{k}f(q^{k}x)\,D_{q}g(q^{k}x)=(1-q)\,x\sum _{k=0}^{\infty }q^{k}f(q^{k}x){\tfrac {g(q^{k}x)-g(q^{k+1}x)}{(1-q)q^{k}x}},}$  or

${\displaystyle \int f(x)\,{\rm {d}}_{q}g(x)=\sum _{k=0}^{\infty }f(q^{k}x)\cdot (g(q^{k}x)-g(q^{k+1}x)),}$ 

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

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 edit

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

• 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 x_n", 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.