Jackson integral

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:

: $\backslash int\_0^a\; f(x)\backslash ,\{\backslash rm\; d\}\_q\; x\; =\; (1-q)\backslash ,a\backslash sum\_\{k=0\}^\{\backslash infty\}q^k\; f(q^k\; a).$

Consistent with this is the definition for $a\; \backslash to\; \backslash infty$

: $\backslash int\_0^\backslash infty\; f(x)\backslash ,\{\backslash rm\; d\}\_q\; x\; =\; (1-q)\backslash sum\_\{k=-\backslash infty\}^\{\backslash infty\}q^k\; f(q^k\; ).$

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

: $\backslash int\; f(x)\backslash ,D\_q\; g\backslash ,\{\backslash rm\; d\}\_q\; x\; =\; (1-q)\backslash ,x\backslash sum\_\{k=0\}^\{\backslash infty\}q^k\; f(q^k\; x)\backslash ,D\_q\; g(q^k\; x)\; =\; (1-q)\backslash ,x\backslash sum\_\{k=0\}^\{\backslash infty\}q^k\; f(q^k\; x)\backslash tfrac\{g(q^\{k\}x)-g(q^\{k+1\}x)\}\{(1-q)q^k\; x\},$ or

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

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 {{Cite journal|last1=Kempf|first1=A|title=Algebraic q-Integration and Fourier Theory on Quantum and Braided Spaces|journal=Journal of Mathematical Physics|volume=35|issue=12|pages=6802–6837|last2=Majid|first2=Shahn|year=1994|arxiv=hep-th/9402037|bibcode=1994JMP....35.6802K|doi=10.1063/1.530644|s2cid=16930694}}).

Suppose that 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.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 x_n", Proc. R. Soc. 74 64–72.
• Jackson F H (1910), "On q-definite integrals", Q. J. Pure Appl. Math. 41 193–203.
• {{cite book |last1=Exton |first1=Harold |title=Q-hypergeometric functions and applications |date=1983 |publisher=E. Horwood |location=Chichester [West Sussex] |isbn=978-0470274538}}

Category:Special functions

Category:Q-analogs

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