, Volume 57, Issue 1, pp 39–66

On discrete q-beta operators

  • Vijay Gupta
  • P. N. Agrawal
  • Durvesh Kumar Verma


The aim of the present paper is to introduce and study the q-analogue of discrete beta operators. First, we show some approximation properties of these operators. Then, we establish some global direct error estimates for the above operators using the second order Ditzian–Totik modulus of smoothness. Finally, we define and study the limit discrete q-beta operator.


q-beta operators q-integer Ditzian–Totik modulus of smoothness 

Mathematics Subject Classification (2000)

41A25 41A30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Altomare, F., Campiti, M.: Korovkin-type approximatiom theory and applications, de Gruyter Studies in Mathematics 17 (ed. H. Bauer, J. L. Kazdan, E. Zehnder), Berlin (1994)Google Scholar
  2. 2.
    Andrews G.E., Askey R., Roy R.: Special functions. Cambridge Universiy Press, Combridge (1999)MATHGoogle Scholar
  3. 3.
    Aral, A., Gupta, V.: Generalized q Baskakov operators, Math. Slovaca, to appearGoogle Scholar
  4. 4.
    Aral A., Gupta V.: On the Durrmeyer type modification of the q-Baskakov type operators. Nonlinear Anal. 72, 1171–1180 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ditzian Z., Totik V.: Moduli of smoothness. Springer, Berlin (1987)MATHGoogle Scholar
  6. 6.
    Ernst, T.: The history of q-calculus and a new method, U. U. D. M. Report 2000, 16. ISSN 1101–3591, Department of Mathematics, Upsala University, (2000)Google Scholar
  7. 7.
    Finta Z., Gupta V.: Approximation properties of q-Baskakov operators. Cent. Eur. J. Math. 8(1), 199–211 (2010)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Gupta V.: Some approximation properties of q-Durrmeyer operators. Appl. Math. Comput. 197(1), 172–178 (2008)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Gupta V., Ahmad A.: Simultaneous approximation by modified Beta operators. Instanbul Uni. Fen. Fak. Mat. Der. 54, 11–22 (1995)MathSciNetGoogle Scholar
  10. 10.
    Kac V., Cheung P.: Quantum calculus, Universitext. Springer, New York (2002)CrossRefGoogle Scholar
  11. 11.
    Kim T.: New approach to q-Euler polynomials of higher order. Russ. J. Math. Phys. 17(2), 218–225 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lupas L.: A property of S. N. Bernstein operators. Mathematica (Cluj) 9(32), 299–301 (1967)MathSciNetGoogle Scholar
  13. 13.
    Lupas L.: On star shapedness preserving properties of a claas of linear positive operators. Mathematica (Cluj) 12(35), 105–109 (1970)MathSciNetGoogle Scholar
  14. 14.
    De Sole A., Kac V.G.: On integral represention of q-gamma and q-beta functions. Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, (9) Mat. Appl. 16(1), 11–29 (2005)MathSciNetMATHGoogle Scholar
  15. 15.
    Pethe S.: On the Baskakov operator. Indian J. Math. 26(1–3), 43–48 (1984)MathSciNetMATHGoogle Scholar
  16. 16.
    Wang H., Meng F.: The rate of convergence of q-Bernstein polynomials for 0 <  q <  1. Approx. Theor. 136, 151–158 (2005)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2011

Authors and Affiliations

  • Vijay Gupta
    • 1
  • P. N. Agrawal
    • 2
  • Durvesh Kumar Verma
    • 2
  1. 1.School of Applied SciencesNetaji Subhash Institute of TechnologyNew DelhiIndia
  2. 2.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia

Personalised recommendations