Q-derivative

The q-derivative of a function f(x) is defined as{{sfn|Jackson|1908|pp=253–281}}{{sfn|Kac|Pokman Cheung|2002}}{{sfn|Ernst|2012}}

:$\backslash left(\backslash frac\{d\}\{dx\}\backslash right)\_q\; f(x)=\backslash frac\{f(qx)-f(x)\}\{qx-x\}.$

It is also often written as $D\_qf(x)$. 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

:$D\_q=\; \backslash frac\{1\}\{x\}\; ~\; \backslash frac\{q^\{d~~~\; \backslash over\; d\; (\backslash ln\; x)\}\; -1\}\{q-1\}\; ~,$

which goes to the plain derivative, $D\_q\; \backslash to\; \backslash frac\{d\}\{dx\}$ as $q\; \backslash to\; 1$.

It is manifestly linear,

:$\backslash displaystyle\; D\_q\; (f(x)+g(x))\; =\; D\_q\; f(x)\; +\; D\_q\; g(x)~.$

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

:$\backslash displaystyle\; D\_q\; (f(x)g(x))\; =\; g(x)D\_q\; f(x)\; +\; f(qx)D\_q\; g(x)\; =\; g(qx)D\_q\; f(x)\; +\; f(x)D\_q\; g(x).$

Similarly, it satisfies a quotient rule,

:$\backslash displaystyle\; D\_q\; (f(x)/g(x))\; =\; \backslash frac\{g(x)D\_q\; f(x)\; -\; f(x)D\_q\; g(x)\}\{g(qx)g(x)\},\backslash quad\; g(x)g(qx)\backslash neq\; 0.$

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

:$\backslash displaystyle\; D\_q\; f(g(x))\; =\; D\_\{q^k\}(f)(g(x))D\_q(g)(x).$

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

Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the monomial is:{{sfn|Kac|Pokman Cheung|2002}}

:$\backslash left(\backslash frac\{d\}\{dz\}\backslash right)\_q\; z^n\; =\; \backslash frac\{1-q^n\}\{1-q\}\; z^\{n-1\}\; =$

[n]_q z^{n-1}

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

The n-th q-derivative of a function may be given as:{{sfn|Ernst|2012}}

:$(D^n\_q\; f)(0)=$

\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

:$\backslash displaystyle\; D\_q(f(x))\; =\; \backslash sum\_\{k=0\}^\{\backslash infty\}\backslash frac\{(q-1)^k\}\{(k+1)!\}\; x^k\; f^\{(k+1)\}(x).$

A q-analog of the Taylor expansion of a function about zero follows:{{sfn|Kac|Pokman Cheung|2002}}

:$f(z)=\backslash sum\_\{n=0\}^\backslash infty\; f^\{(n)\}(0)\backslash ,\backslash frac\{z^n\}\{n!\}\; =\; \backslash sum\_\{n=0\}^\backslash infty\; (D^n\_q\; f)(0)\backslash ,\backslash frac\{z^n\}\{[n]!\_q\}.$

The following representation for higher order $q$-derivatives is known:{{sfn|Koepf|2014}}{{sfn|Koepf|Rajković|Marinković|2007|pp=621–638}}

:$D\_q^nf(x)=\backslash frac\{1\}\{(1-q)^nx^n\}\backslash sum\_\{k=0\}^n(-1)^k\backslash binom\{n\}\{k\}\_q\; q^\{\backslash binom\{k\}\{2\}-(n-1)k\}f(q^kx).$

$\backslash binom\{n\}\{k\}\_q$ is the $q$-binomial coefficient. By changing the order of summation as $r=n-k$, we obtain the next formula:{{sfn|Koepf|2014}}{{sfn|Annaby|Mansour|2008|pp=472–483}}

:$D\_q^nf(x)=\backslash frac\{(-1)^n\; q^\{-\backslash binom\{n\}\{2\}\}\}\{(1-q)^nx^n\}\backslash sum\_\{r=0\}^n(-1)^r\backslash binom\{n\}\{r\}\_q\; q^\{\backslash binom\{r\}\{2\}\}f(q^\{n-r\}x).$

Higher order $q$-derivatives are used to $q$-Taylor formula and the $q$-Rodrigues' formula (the formula used to construct $q$-orthogonal polynomials{{sfn|Koepf|2014}}).

Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator:Gupta V., Rassias T.M., Agrawal P.N., Acu A.M. (2018) Basics of Post-Quantum Calculus. In: Recent Advances in Constructive Approximation Theory. SpringerOptimization and Its Applications, vol 138. Springer.{{sfn|Duran|2016}}

:$D\_\{p,q\}f(x):=\backslash frac\{f(px)-f(qx)\}\{(p-q)x\},\backslash quad\; x\backslash neq\; 0.$

Wolfgang Hahn introduced the following operator (Hahn difference):Hahn, W. (1949). Math. Nachr. 2: 4-34.Hahn, W. (1983) Monatshefte Math. 95: 19-24.

:

When $\backslash omega\backslash to0$ this operator reduces to $q$-derivative, and when $q\backslash to1$ it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems.{{sfn|Foupouagnigni|1998}}Kwon, K.; Lee, D.; Park, S.; Yoo, B.: Kyungpook Math. J. 38, 259-281 (1998).Alvarez-Nodarse, R.: J. Comput. Appl. Math. 196, 320-337 (2006).

$\backslash beta$-derivative is an operator defined as follows:Auch, T. (2013): Development and Application of Difference and Fractional Calculus on Discrete Time Scales. PhD thesis, University of Nebraska-Lincoln.{{sfn|Hamza|Sarhan|Shehata|Aldwoah|2015|p=182}}

:$D\_\backslash beta\; f(t):=\backslash frac\{f(\backslash beta(t))-f(t)\}\{\backslash beta(t)-t\},\backslash quad\backslash beta\backslash neq\; t,\backslash quad\backslash beta:I\backslash to\; I.$

In the definition, $I$ is a given interval, and $\backslash beta(t)$ is any continuous function that strictly monotonically increases (i.e. $t>s\backslash rightarrow\backslash beta(t)>\backslash beta(s)$). When $\backslash beta(t)=qt$ then this operator is $q$-derivative, and when $\backslash beta(t)=qt+\backslash omega$ this operator is Hahn difference.

The q-calculus has been used in machine learning for designing stochastic activation functions.{{sfn|Nielsen|Sun|2021|pp=2782–2789}}

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

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{{refend}}

Category:Differential calculus

Category:Generalizations of the derivative

Category:Linear operators in calculus

Category:Q-analogs