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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

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

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

It is manifestly linear,

$\displaystyle D_{q}(f(x)+g(x))=D_{q}f(x)+D_{q}g(x)~.$

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

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

Similarly, it satisfies a quotient rule,

$\displaystyle D_{q}(f(x)/g(x))={\frac {g(x)D_{q}f(x)-f(x)D_{q}g(x)}{g(qx)g(x)}},\quad g(x)g(qx)\neq 0.$

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

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

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 $\lim _{q\to 1}[n]_{q}=n$ 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

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

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

$f(z)=\sum _{n=0}^{\infty }f^{(n)}(0)\,{\frac {z^{n}}{n!}}=\sum _{n=0}^{\infty }(D_{q}^{n}f)(0)\,{\frac {z^{n}}{[n]_{q}!}}$

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),