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-derivativeedit
-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.
Applicationsedit
The q-calculus has been used in machine learning for designing stochastic activation functions.[16]
^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.
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. CiteSeerX10.1.1.298.4595. 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.