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Template:Lowercase In mathematics, in the area of combinatorics, 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-integration

Definition

The q-derivative of a function f(x) is defined as

${\left(\frac{d}{dx}\right)}_{q}f(x)=\frac{f(qx)-f(x)}{qx-x}.$

It is also often written as $D_{q}f(x)$. The q-derivative is also known as the Jackson derivative.

Formally, in terms of Lagrange's shift operator in logarithmic variables, it amounts to the operator

$D_q=\frac{1}{x}\frac{q^{\frac{d}{d(\ln x)}}-1}{q-1},$

which goes to the plain derivative, →^d⁄_dx, as q→1.

It is manifestly linear,

$D_q(f(x)+g(x))=D_{q}f(x)+D_{q}g(x).$

It has product rule analogous to the ordinary derivative product rule, with two equivalent forms

$D_q(f(x)g(x))=g(x)D_{q}f(x)+f(qx)D_{q}g(x)=g(qx)D_{q}f(x)+f(x)D_{q}g(x).$

Similarly, it satisfies a quotient rule,

$D_q(f(x)/g(x))=\frac{g(x)D_{q}f(x)-f(x)D_{q}g(x)}{g(qx)g(x)},g(x)g(qx)\ne 0.$

There is also a rule similar to the chain rule for ordinary derivatives. Let $g(x)=c{x}^k$. Then

$D_{q}f(g(x))=D_{q^k}(f)(g(x))D_q(g)(x).$

The eigenfunction of the q-derivative is the q-exponential e_q(x).

Relationship to ordinary derivatives

Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the monomial is:

${\left(\frac{d}{dz}\right)}_q{z}^n=\frac{1-q^n}{1-q}{z}^{n-1}=[n{]}_q{z}^{n-1}$

where $[n{]}_q$ is the q-bracket of n. Note that $\underset{q\to 1}{\lim }[n{]}_q=n$ so the ordinary derivative is regained in this limit.

The n-th q-derivative of a function may be given as:

$(D_q^{n}f)(0)=\frac{f^{(n)}(0)}{n!}\frac{(q;q{)}_n}{(1-q{)}^n}=\frac{f^{(n)}(0)}{n!}[n{]}_q!$

provided that the ordinary n-th derivative of f exists at x=0. Here, $(q;q{)}_n$ is the q-Pochhammer symbol, and $[n{]}_q!$ is the q-factorial. If $f(x)$ is analytic we can apply the Taylor formula to the definition of $D_q(f(x))$ to get

$D_q(f(x))=\sum _{k=0}^{\mathrm{\infty }}\frac{(q-1{)}^k}{(k+1)!}{x}^k{f}^{(k+1)}(x).$

A q-analog of the Taylor expansion of a function about zero follows:

$f(z)=\sum _{n=0}^{\mathrm{\infty }}{f}^{(n)}(0)\frac{z^n}{n!}=\sum _{n=0}^{\mathrm{\infty }}(D_q^{n}f)(0)\frac{z^n}{[n{]}_q!}$

See also

• Derivative (generalizations)
• Jackson integral
• Q-exponential
• Q-difference polynomials
• Quantum calculus
• Tsallis entropy

References

• F. H. Jackson (1908), "On q-functions and a certain difference operator", Trans. Roy. Soc. Edin., 46 253-281.

• Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538

• Victor Kac, Pokman Cheung, Quantum Calculus, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8

Further reading

• J. Koekoek, R. Koekoek, A note on the q-derivative operator, (1999) ArXiv math/9908140
• Thomas Ernst, The History of q-Calculus and a new method,(2001),