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Mathematica Slovaca

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Volume 68, Issue 4


Approximation by Baskakov-Durrmeyer operators based on (p, q)-integers

Tuncer Acar / Ali Aral / Mohammad Mursaleen
Published Online: 2018-08-06 | DOI: https://doi.org/10.1515/ms-2017-0153


In the present paper, we introduce a new sequence of linear positive operators based on (p, q)-integers. To approximate functions over unbounded intervals, we introduce Baskakov-Durrmeyer type operators using the (p, q)-Gamma function. We investigate rate of convergence of new operators in terms of modulus of continuities and obtain their approximation behavior for the functions belonging to Lipschitz class. At the end, we present a modification of new operators preserving the test function x.

MSC 2010: Primary 41A35; Secondary 41A25

Keywords: (p, q)-integers; (p, q)-Gamma function; (p, q)-Baskakov-Durrmeyer operators; rate of convergence


  • [1]

    Acar, T.: (p, q)-generalization of Szász-Mirakyan operators, Math. Methods Appl. Sci. 39(10) (2016), 2685–2695.CrossrefGoogle Scholar

  • [2]

    Acar, T.—Aral, A.—Mohiuddine, S. A.: Approximation by bivariate (p, q)-Bernstein-Kantorovich operators, Iran. J. Sci. Technol. Trans. A Sci. (2016), .CrossrefGoogle Scholar

  • [3]

    Acar, T.—Aral, A.—Mohiuddine, S. A.: On Kantorovich modifications of (p, q)-Baskakov operators, J. Inequal. Appl. (2016), 2016: 98.CrossrefGoogle Scholar

  • [4]

    Acar, T.—Agrawal, P. N.—Kumar, S.: On a modification of (p, q)-Szasz-Mirakyan operators, Comp. Anal. Oper. Theo. 12 (2018), 155–167.CrossrefGoogle Scholar

  • [5]

    Aral, A.—Gupta, V.: Generalized q-Baskakov operators, Math. Slovaca 61 (2011), 619–634.Web of ScienceGoogle Scholar

  • [6]

    Aral, A.—Gupta, V.: Applications of (p, q)-gamma function to Szasz-Durrmeyer operators, Publications de ľInstitut Mathematique, In Press.Google Scholar

  • [7]

    Aral, A.—Gupta, V.: (p, q)-Type beta functions of second kind, Advances in Operator Theory 1(1) (2016), 134–146.Web of ScienceGoogle Scholar

  • [8]

    Burban, I.: Two-parameter deformation of the oscillator albegra and (p, q) analog of two dimensional conformal field theory, J. Nonlinear Math. Phys. 2(3–4) (1995), 384–391.Google Scholar

  • [9]

    Burban, I.—Klimyk, A. U.: P, Q differentiation, P, Q integration and P, Q hypergeometric functions related to quantum groups, Integral Transforms Spec. Funct. 2(1) (1994), 15–36.CrossrefGoogle Scholar

  • [10]

    Devore, R. A.—Lorentz G. G.: Constructive Approximation, Springer, Berlin, 1993.Google Scholar

  • [11]

    Hounkonnou, M. N.—Desire, J.—Kyemba, B.: 𝓡(p, q)-calculus: differentiation and integration, SUT J. Math. 49(2) (2013), 145–167.Google Scholar

  • [12]

    Ibikli, E.—Gadjieva, E. A.: The order of approximation of some unbounded function by the sequences of positive linear operators, Turk. J. Math. 19(3) (1995), 331–337.Google Scholar

  • [13]

    Jagannathan, R.—Rao, K. S.: Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series. In: Proceedings of the International Conference on Number Theory and Mathematical Physics, 20–21 December 2005.Google Scholar

  • [14]

    King, J. P.: Positive linear operators which preserves x2, Acta. Math. Hungar. 99 (2003), 203–208.CrossrefGoogle Scholar

  • [15]

    Lenze, B.: On Lipschitz type maximal functions and their smoothness spaces, Indag. Math. 50 (1988), 53–63.Google Scholar

  • [16]

    Mursaleen, M.—Ansari, K. J.—Khan, A.: On (p, q)-analogue of Bernstein operators, Appl. Math. Comput. 266 (2015), 874–882.Web of ScienceGoogle Scholar

  • [17]

    Mursaleen, M.—Ansari, K. J.—Khan, A.: Some approximation results by (p, q)-analogue of Bernstein-Stancu operators, Appl. Math. Comput. 264 (2015), 392–402 [Corrigendum: Appl. Math. Comput. 269 (2015), 744–746].Web of ScienceGoogle Scholar

  • [18]

    Mursaleen, M.—Nasiruzzaman, Md.—Khan, A.—Ansari, K. J.: Some approximation results on Bleimann-Butzer-Hahn operators defined by (p, q)-integers, Filomat 30(3) (2016), 639–648.CrossrefWeb of ScienceGoogle Scholar

  • [19]

    Mursaleen, M.—Nasiruzzaman, Md.—Nurgali, A.: Some approximation results on Bernstein-Schurer operators defined by (p, q)-integers, J. Ineq. Appl. 2015 (2015), 249.CrossrefGoogle Scholar

  • [20]

    Sadjang, P. N.: On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas, arXiv:1309.3934 [math.QA].Google Scholar

  • [21]

    Sahai, V.—Yadav, S.: Representations of two parameter quantum algebras and p, q-special functions, J. Math. Anal. Appl. 335 (2007), 268–279.Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2016-09-26

Accepted: 2017-03-08

Published Online: 2018-08-06

Published in Print: 2018-08-28

Communicated by Gregor Dolinar

Citation Information: Mathematica Slovaca, Volume 68, Issue 4, Pages 897–906, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0153.

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