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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 Chung et al. (1994).

Definition

The q-derivative of a function f(x) is defined as^[1]^[2]^[3]

\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, $D_{q}\to {\frac {d}{dx}}$ as $q\to 1$ .

It is manifestly linear,

\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

\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:^[2]

\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:^[3]

(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:^[2]

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:^[4]^[5]

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:^[4]^[6]

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^[4]).

Generalizations

Post Quantum Calculus

Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator:^[7]^[8]

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

Hahn difference

Wolfgang Hahn introduced the following operator (Hahn difference):^[9]^[10]

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

When $\omega \to 0$ 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.^[11]^[12]^[13]

β-derivative

$\beta$ -derivative is an operator defined as follows:^[14]^[15]

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

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

Applications

The q-calculus has been used in machine learning for designing stochastic activation functions.^[16]

Citations

^ Jackson 1908, pp. 253–281.
^ ^a ^b ^c Kac & Pokman Cheung 2002.
^ ^a ^b Ernst 2012.
^ ^a ^b ^c Koepf 2014.
^ Koepf, Rajković & Marinković 2007, pp. 621–638.
^ Annaby & Mansour 2008, pp. 472–483.
^ 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.
^ Duran 2016.
^ Hahn, W. (1949). Math. Nachr. 2: 4-34.
^ Hahn, W. (1983) Monatshefte Math. 95: 19-24.
^ 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).
^ Auch, T. (2013): Development and Application of Difference and Fractional Calculus on Discrete Time Scales. PhD thesis, University of Nebraska-Lincoln.
^ Hamza et al. 2015, p. 182.
^ Nielsen & Sun 2021, pp. 2782–2789.

Bibliography

Annaby, M. H.; Mansour, Z. S. (2008). "q-Taylor and interpolation-difference operators". Journal of Mathematical Analysis and Applications. 344 (1): 472–483. doi:10.1016/j.jmaa.2008.02.033.
Chung, K. S.; Chung, W. S.; Nam, S. T.; Kang, H. J. (1994). "New q-derivative and q-logarithm". International Journal of Theoretical Physics. 33 (10): 2019–2029. Bibcode:1994IJTP...33.2019C. doi:10.1007/BF00675167. S2CID 117685233.
Duran, U. (2016). Post Quantum Calculus (M.Sc. thesis). Department of Mathematics, University of Gaziantep Graduate School of Natural & Applied Sciences. Retrieved 9 March 2022 – via ResearchGate.
Ernst, T. (2012). A comprehensive treatment of q-calculus. Springer Science & Business Media. ISBN 978-303480430-1.
Ernst, Thomas (2001). "The History of q-Calculus and a new method" (PDF). Archived from the original (PDF) on 28 November 2009. Retrieved 9 March 2022.
Exton, H. (1983). q-Hypergeometric Functions and Applications. New York: Halstead Press. ISBN 978-047027453-8.
Foupouagnigni, M. (1998). Laguerre-Hahn orthogonal polynomials with respect to the Hahn operator: fourth-order difference equation for the rth associated and the Laguerre-Freud equations for the recurrence coefficients (Ph.D. thesis). Université Nationale du Bénin.
Hamza, A.; Sarhan, A.; Shehata, E.; Aldwoah, K. (2015). "A General Quantum Difference Calculus". Advances in Difference Equations. 1: 182. doi:10.1186/s13662-015-0518-3. S2CID 54790288.
Jackson, F. H. (1908). "On q-functions and a certain difference operator". Trans. R. Soc. Edinb. 46 (2): 253–281. doi:10.1017/S0080456800002751. S2CID 123927312.
Kac, Victor; Pokman Cheung (2002). Quantum Calculus. Springer-Verlag. ISBN 0-387-95341-8.
Koekoek, J.; Koekoek, R. (1999). "A note on the q-derivative operator". J. Math. Anal. Appl. 176 (2): 627–634. arXiv:math/9908140. doi:10.1006/jmaa.1993.1237. S2CID 329394.
Koepf, W.; Rajković, P. M.; Marinković, S. D. (July 2007). "Properties of q-holonomic functions". Journal of Difference Equations and Applications. 13 (7): 621–638. CiteSeerX 10.1.1.298.4595. doi:10.1080/10236190701264925. S2CID 123079843.
Koepf, Wolfram (2014). Hypergeometric Summation. An Algorithmic Approach to Summation and Special Function Identities. Springer. ISBN 978-1-4471-6464-7.
Nielsen, Frank; Sun, Ke (2021). "q-Neurons: Neuron Activations Based on Stochastic Jackson's Derivative Operators". IEEE Trans. Neural Netw. Learn. Syst. 32 (6): 2782–2789. arXiv:1806.00149. doi:10.1109/TNNLS.2020.3005167. PMID 32886614. S2CID 44143912.

[FOOTNOTEJackson1908253–281-1] Jackson 1908, pp. 253–281.

[FOOTNOTEKacPokman_Cheung2002-2] Kac & Pokman Cheung 2002.

[FOOTNOTEErnst2012-3] Ernst 2012.

[FOOTNOTEKoepf2014-4] Koepf 2014.

[FOOTNOTEKoepfRajkovićMarinković2007621–638-5] Koepf, Rajković & Marinković 2007, pp. 621–638.

[FOOTNOTEAnnabyMansour2008472–483-6] Annaby & Mansour 2008, pp. 472–483.

[7] 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.

[FOOTNOTEDuran2016-8] Duran 2016.

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

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

[FOOTNOTEFoupouagnigni1998-11] Foupouagnigni 1998.

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

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

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

[FOOTNOTEHamzaSarhanShehataAldwoah2015182-15] Hamza et al. 2015, p. 182.

[FOOTNOTENielsenSun20212782–2789-16] Nielsen & Sun 2021, pp. 2782–2789.

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