q-derivative

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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[edit]

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

It is also often written as . 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

which goes to the plain derivative as .

It is manifestly linear,

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

Similarly, it satisfies a quotient rule,

There is also a rule similar to the chain rule for ordinary derivatives. Let . Then

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

Relationship to ordinary derivatives[edit]

Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the monomial is:[2]

where is the q-bracket of n. Note that so the ordinary derivative is regained in this limit.

The n-th q-derivative of a function may be given as:[3]

provided that the ordinary n-th derivative of f exists at x = 0. Here, is the q-Pochhammer symbol, and is the q-factorial. If is analytic we can apply the Taylor formula to the definition of to get

A q-analog of the Taylor expansion of a function about zero follows:[2]

Higher order q-derivatives[edit]

Th following representation for higher order -derivatives is known:[4][5]

is the -binomial coefficient. By changing the order of summation as , we obtain the next formula:[4][6]

Higher order -derivatives are used to -Taylor formula and the -Rodrigues' formula (the formula used to construct -orthogonal polynomials[4]).

Generalizations[edit]

Post Quantum Calculus[edit]

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

Hahn difference[edit]

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

When this operator reduces to -derivative, and when 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[edit]

-derivative is an operator defined as follows:[14][15]

In the definition, is a given interval, and is any continuous function that strictly monotonically increases (i.e. ). When then this operator is -derivative, and when this operator is Hahn difference.

Applications[edit]

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

See also[edit]

Citations[edit]

  1. ^ Jackson 1908, pp. 253–281.
  2. ^ a b c Kac & Pokman Cheung 2002.
  3. ^ a b Ernst 2012.
  4. ^ a b c Koepf 2014.
  5. ^ Koepf, Rajković & Marinković 2007, pp. 621–638.
  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.
  8. ^ Duran 2016.
  9. ^ Hahn, W. (1949). Math. Nachr. 2: 4-34.
  10. ^ Hahn, W. (1983) Monatshefte Math. 95: 19-24.
  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.
  15. ^ Hamza et al. 2015, p. 182.
  16. ^ Nielsen & Sun 2021, pp. 2782–2789.

Bibliography[edit]

  • 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. 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.
  • 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. 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 Networks Learn. Syst. 32 (6): 2782–2789. arXiv:1806.00149. doi:10.1109/TNNLS.2020.3005167. PMID 32886614. S2CID 44143912.