q-analog In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q''- ...

theory, the Jackson integral

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...

in the theory of

special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...

that expresses the operation inverse to

q-differentiation In mathematics, in the area of combinatorics and quantum calculus, the ''q''-derivative, or Jackson derivative, is a ''q''-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's ''q''-integrati ...

. The Jackson integral was introduced by

Frank Hilton Jackson The Reverend Frank Hilton Jackson (16 August 1870, Hull, England – 27 April 1960) was an English clergyman and mathematician who worked on basic hypergeometric series. He introduced several ''q''-analogs such as the Jackson–Bessel functions ...

. For methods of numerical evaluation, see and .

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: :

\int_0^a f(x)\,_q x = (1-q)\,a\sum_^q^k f(q^k a).

Consistent with this is the definition for

a \to \infty

\int_0^\infty f(x)\,_q x = (1-q)\sum_^q^k f(q^k ).

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

\int f(x)\,D_q g\,_q x = (1-q)\,x\sum_^q^k f(q^k x)\,D_q g(q^k x) = (1-q)\,x\sum_^q^k f(q^k x)\tfrac,

or :

\int f(x)\,_q g(x) = \sum_^ f(q^k x)\cdot(g(q^x)-g(q^x)),

giving a ''q''-analogue of the

Riemann–Stieltjes integral In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...

Jackson integral as q-antiderivative

Just as the ordinary

antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...

of a

continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...

can be represented by its

Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...

, 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 If, f(x)x^\alpha, is bounded on the interval

Quantum Calculus Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus", where h ostensibly stands for Planck's constant while ''q'' stan ...

'', Universitext, Springer-Verlag, 2002. *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. * Special functions Q-analogs {{mathanalysis-stub