On the Fundamental Theorem of $$\varvec{(p,q)}$$ -Calculus and Some $$\varvec{(p,q)}$$ -Taylor Formulas
Tóm tắt
In this paper, the (p, q)-derivative and the (p, q)-integration are investigated. Two suitable polynomial bases for the (p, q)-derivative are provided and various properties of these bases are given. As application, two (p, q)-Taylor formulas for polynomials are given, the fundamental theorem of (p, q)-calculus is included and the formula of (p, q)-integration by part is proved.
Tài liệu tham khảo
Acar, T.: \((p, q)\)-generalization of Szász-Mirakyan operators. Math. Methods Appl. Sci. 39, 2685 (2015)
Acar, T., Aral, A., Mohiuddine, S.A.: Approximation by bivariate \((p, q)\)-Bernstein–Kantorovich operators. Iran. J. Sci. Technol. Trans. A Sci (2016). https://doi.org/10.1007/s40995-016-0045-4
Acar, T., Aral, A., Mohiuddine, S.A.: On Kantorovich modification of \((p, q)\)-Baskakov operators. J. Inequal. Appl. 98, 98 (2016)
Bukweli-Kyemba, J.D., Hounkonnou, M.N.: Quantum deformed algebras: coherent states and special functions (2013). arXiv:1301.0116v1
Burban, I.M., Klimyk, A.U.: \((P, Q)\)-Differentiation, \((P, Q)\) integration, and \((P, Q)\)-hypergeometric functions related to quantum groups. Integral Transform Spec. Funct. 2(1), 15–36 (1994)
Chakrabarti, R., Jagannathan, R.: A \((p, q)\)-oscillator realization of two-parameter quantum algebras. J. Phys. A Math. Gen. 24, L711 (1991)
Chung, W.S., Jung, M.: Some new properties concerning the Q-deformed calculus and the Q-deformation of the diffusion equation. Mod. Phys. Lett. A 28, 1350100 (2013)
Diaz, R.: Feynman–Jackson integrals. J. Nonlinear Math. Phys. 13, 365 (2006). https://arxiv.org/abs/math/0508395
Ernst, T.: The history of \(q\)-calculus and a new method, (Licentiate Thesis), U.U.D.M. Report 2000: 16. http://www.math.uu.se/thomas/Lics.pdf. Accessed 2 March 2001
Fiore, G.: On the hermiticity of \(q\)-differential operators and forms on the quantum euclidean spaces \(R^{**}N(q)\). Rev. Math. Phys. 18, 79 (2006). arxiv:math/0403463 [math-qa]
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Applied and Computational Mathematics, vol. 35. Cambridge University Press, Cambridge (1990)
Jagannathan, R.: \((P,Q)\)-Special Functions (1998). arXiv:math/9803142v1
Jagannathan, R., Srinivasa Rao, K.: Tow-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series (2006). arXiv:math/0602613v
Kac, V., Cheung, P.: Quantum Calculus. Springer, Berlin (2001)
Njionou Sadjang, P.: On two \((p, q)\)-analogues of the Laplace transform. J. Difference Eqn. Appl. 23(9), 1562–1583
Wachter, H.: \(q\)-integration on quantum spaces. Eur. Phys. J. C 32, 281 (2004). arXiv:hep-th/0206083