q-derivative

Template:Short description

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-integration. For other forms of q-derivative, see Template:Harvtxt.

Definition

The q-derivative of a function f(x) is defined asTemplate:Sfn Template:Sfn Template:Sfn

{(\frac{d}{d x})}_{q} f (x) = \frac{f (q x) - f (x)}{q x - 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, $D_{q} \to \frac{d}{d x}$ as $q \to 1$ .

It is manifestly linear,

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

D_{q} (f (x) g (x)) = g (x) D_{q} f (x) + f (q x) D_{q} g (x) = g (q x) 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 (q x) g (x)}, g (x) g (q x) \neq 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:Template:Sfn

{(\frac{d}{d z})}_{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:Template:Sfn

(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}^{\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:Template:Sfn

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}} .

Higher order q-derivatives

The following representation for higher order $q$ -derivatives is known:Template:Sfn Template:Sfn

D_{q}^{n} f (x) = \frac{1}{(1 - q)^{n} x^{n}} \sum_{k = 0}^{n} (- 1)^{k} {(\binom{n}{k})}_{q} q^{(\binom{k}{2}) - (n - 1) k} f (q^{k} x) .

${(\binom{n}{k})}_{q}$ is the $q$ -binomial coefficient. By changing the order of summation as $r = n - k$ , we obtain the next formula:Template:Sfn Template:Sfn

D_{q}^{n} f (x) = \frac{(- 1)^{n} q^{- (\binom{n}{2})}}{(1 - q)^{n} x^{n}} \sum_{r = 0}^{n} (- 1)^{r} {(\binom{n}{r})}_{q} q^{(\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 Template:Sfn).

Generalizations

Post Quantum Calculus

Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator:^[1]Template:Sfn

D_{p, q} f (x) : = \frac{f (p x) - f (q x)}{(p - q) x}, x \neq 0 .

Hahn difference

Wolfgang Hahn introduced the following operator (Hahn difference):^[2]^[3]

D_{q, ω} f (x) : = \frac{f (q x + ω) - f (x)}{(q - 1) x + ω}, 0 < q < 1, ω > 0 .

When $ω \to 0$ this operator reduces to $q$ -derivative, and when $q \to 1$ it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems.Template:Sfn^[4]^[5]

β-derivative

$β$ -derivative is an operator defined as follows:^[6]Template:Sfn

D_{β} f (t) : = \frac{f (β (t)) - f (t)}{β (t) - t}, β \neq t, β : I \to I .

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

Applications

The q-calculus has been used in machine learning for designing stochastic activation functions.Template:Sfn

Citations

Template:Reflist

Bibliography

Template:Refbegin

Template:Refend

↑ 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.
↑ Hahn, W. (1949). Math. Nachr. 2: 4-34.
↑ Hahn, W. (1983) Monatshefte Math. 95: 19-24.
↑ 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).
↑ Auch, T. (2013): Development and Application of Difference and Fractional Calculus on Discrete Time Scales. PhD thesis, University of Nebraska-Lincoln.

[1] 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.

[2] Hahn, W. (1949). Math. Nachr. 2: 4-34.

[3] Hahn, W. (1983) Monatshefte Math. 95: 19-24.

[4] Kwon, K.; Lee, D.; Park, S.; Yoo, B.: Kyungpook Math. J. 38, 259-281 (1998).

[5] Alvarez-Nodarse, R.: J. Comput. Appl. Math. 196, 320-337 (2006).

[6] Auch, T. (2013): Development and Application of Difference and Fractional Calculus on Discrete Time Scales. PhD thesis, University of Nebraska-Lincoln.

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q-derivative

Contents

Definition

Relationship to ordinary derivatives

Higher order q-derivatives

Generalizations

Post Quantum Calculus

Hahn difference

β-derivative

Applications

See also

Citations

Bibliography

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q-derivative

Definition

Relationship to ordinary derivatives

Higher order q-derivatives

Generalizations

Post Quantum Calculus

Hahn difference

β-derivative

Applications

See also

Citations

Bibliography

Navigation menu

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