Complex Analysis and Operator Theory

, Volume 10, Issue 8, pp 1725–1740 | Cite as

Approximation by (pq)-Lorentz Polynomials on a Compact Disk

  • M. Mursaleen
  • Faisal Khan
  • Asif Khan


The (pq)-factors were introduced in order to generalize or unify several forms of q-oscillator algebras well known in the physics literature related to the representation theory of single parameter quantum algebras. This notion has been recently used in approximation by positive linear operators via (pq)-calculus which has emerged a very active area of research. In this paper, we introduce a new analogue of Lorentz polynomials based on (pq)-integers. We obtain quantitative estimate in the Voronovskaja’s type theorem and exact orders in simultaneous approximation by the complex (pq)-Lorentz polynomials of degree \(n\in \mathbb {N}\) (\(q>p>1)\), attached to analytic functions on compact disks of the complex plane. In this way, we put in evidence the overconvergence phenomenon for the (pq)-Lorentz polynomial, namely the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane.


(p, q)-Integer (p, q)-Lorentz polynomial Voronovskaja’s theorem Iterates Compact disk 

Mathematics Subject Classification

41A10 41A25 41A36 



Authors are very much thankful to the learned referee for his/ her valuable comments which improved the presentation of the paper.


  1. 1.
    Acar, T.: \((p, q)\)-generalization of Szász-Mirakyan operators. Math. Methods Appl. Sci. (2015). doi: 10.1002/mma.3721 MathSciNetzbMATHGoogle Scholar
  2. 2.
    Acar, T., Aral, A., Mohiuddine, S.A.: On Kantorovich modification of \((p, q)\)-Baskakov operators. J. Inequal. Appl. 2016, 98 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burban, I.: Two-parameter deformation of the oscillator albegra and \(\left( p, q\right) \) analog of two dimensional conformal field theory. Nonlinear Math. Phys. 2(3–4), 384–391 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caia, Q.-B., Zhou, G.: On \((p, q)\)-analogue of Kantorovich type Bernstein–Stancu–Schurer operators. Appl. Math. Comput. 276, 12–20 (2016)MathSciNetGoogle Scholar
  5. 5.
    Gal, S.G., Mahmudov, N.I., Kara, M.: Approximation by complex \(q\)-Szász-Kantorovich operators in compact disks. Complex Anal. Oper. Theory. doi: 10.1007/s11785-012-0257-3
  6. 6.
    Gal, S.G.: Overconvergence in Complex Approximtion. Springer, New York (2013)CrossRefGoogle Scholar
  7. 7.
    Gal, S.G.: Approximtion by complex \(q\). Mathematica (Cluj) 54(77), 53–63 (2012)zbMATHGoogle Scholar
  8. 8.
    Hounkonnou, M.N., Désiré, J., Kyemba, B.: \({\cal R} (p, q)\)-calculus: differentiation and integration. SUT J. Math. 49(2), 145–167 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gupta, V.: \((p,q)\)-Szász–Mirakyan–Baskakov operators. Complex Anal. Oper. Theory. doi: 10.1007/s11785-015-0521-4
  10. 10.
    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, pp. 20–21 (2006)Google Scholar
  11. 11.
    Katriel, J., Kibler, M.: Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers. J. Phys. A Math. Gen. 25, 2683–2691 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lorentz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea Publication, New York (1986)zbMATHGoogle Scholar
  13. 13.
    Mahmudov, N.I.: Convergence properties and iterations for q-Stancu polynomials in compact disks. Comput. Math. Appl. 59(12), 3763–3769 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mahmudov, N.I.: Approximation properties of complex q-Szá sz–Mirakjan operators in compact disks. Comput. Math. Appl. 60, 1784–1791 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mahmudov, N.I., Kara, M.: Approximation theorems for generalized complex Kantorovich-type operators. J. Appl. Math. 2012, Article ID 454579. doi: 10.1155/2012/454579
  16. 16.
    Mursaleen, M., Alotaibi, A., Ansari, K.J.: On a Kantorovich variant of \((p,q)\)-Szász–Mirakjan operators. J. Funct. Spaces 2016, Article ID 1035253Google Scholar
  17. 17.
    Mursaleen, M., Ansari, K.J., Khan, A.: Some approximation results by \((p,q)\)-analogue of Bernstein-Stancu operators. Appl. Math. Comput. 264, 392–402 (2015) [Corrigendum: Appl. Math. Comput. 269, 744–746 (2015)]Google Scholar
  18. 18.
    Mursaleen, M., Ansari, K.J., Khan, A.: On \((p,q)\)-analogue of Bernstein operators. Appl. Math. Comput. 266, 874–882 (2015) [Erratum: Appl. Math. Comput. 278, 70–71 (2016)]Google Scholar
  19. 19.
    Mursaleen, M., Nasiruzzaman, Md., Khan, A., Ansari, K.J.: Some approximation results on Bleimann-Butzer-Hahn operators defined by \( (p,q)\)-integers. Filomat (to appear)Google Scholar
  20. 20.
    Mursaleen, M., Nasiuzzaman, Md, Nurgali, A.: Some approximation results on Bernstein–Schurer operators defined by \((p, q)\) -integers. J. Inequal. Appl. 2015, 249 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sharma, H., Gupta, C.: On \((p, q)\)-generalization of Szá sz–Mirakyan Kantorovich operators. Boll. Unione Mat. Ital. 8(3), 213–222 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sadjang, P.N.: On the fundamental theorem of \((p,q)\)-Taylor formulas. arXiv:1309.3934 [math.QA]
  23. 23.
    Sahai, V., Yadav, S.: Representations of two parameter quantum algebras and \(p\),\(q\)-special functions. J. Math. Anal. Appl. 335, 268–279 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia

Personalised recommendations