Asymptotic approximation with Stancu Beta operators


  • Ulrich Abel Wettenberg, Germany


Stancu beta operators, asymptotic approximation, asymptotic expansion, Stirling numbers of the first and second kind, reciprocal factorials
Abstract views: 196


The concern of this paper is a beta type operator \(L_n\) recently introduced by D.D. Stancu.
We present the complete asymptotic expansion for \(L_n\) as \(n\) tends to infinity.
All coefficients of \(n^{-k} \ (k=1,2, \ldots)\) are calculated explicitly in terms of Stirling numbers of the first and second kind.
Moreover, we give an asymptotic expansion for \(L_n\) into a series of reciprocal factorials.


Download data is not yet available.


U. Abel, The moments for the Meyer-König and Zeller operators, J. Approx. Theory, 82 (1995), pp. 352-361,

U. Abel, On the asymptotic approximation with operators of Bleimann, Butzer and Hahn, Indag. Math., New Ser., 7 (1996), pp. 1-9,

U. Abel, The complete asymptotic expansion for the Meyer-König and Zeller operators, J. Math. Anal. Appl., 208 (1997), pp. 109-119,

U. Abel, The asymptotic expansion for the Stancu operators, submitted.

U. Abel, Asymptotic approximation with Kantorovich polynomials, submitted.

J. A. Adell and J. de la Cal, On a Bernstein-type operator associated with the inverse Polya-Eggenberger distribution, Rend. Circolo Matem. Palermo, Ser. II, Nr. 33 (1993), pp. 143-154.

F. Altomare and M. Campiti, "Korovkin-type approximation theory and its applications," Walter de Gruyter, Berlin, New York, 1994,

S. N. Bernstein, Complément à l'article de E. Voronowskaja, Dokl. Akad. Nauk USSR, 4 (1932), pp. 86-92.

R. A. DeVore and G. G. Lorentz, "Constructive approximation", Springer, Berlin, Heidelberg 1993,

C. Jordan, "Calculus of finite differences," Chelsea, New York, 1965.

M. K. Khan, Approximation properties of beta operators, Progress in Approx. Theory, Academic Press, New York, 1991, pp. 483-495.

G. G. Lorentz, "Bernstein polynomials," University of Toronto Press, Toronto, 1953.

A. Lupaş, "Die Folge der Beta-Operatoren," Dissertation, Universität Stuttgart, 1972.

G. Mühlbach, Verallgemeinerungen der Bernstein-und der Lagrange-Polynome. Bemerkungen zu einer Klasse linearer Polynomoperatoren von D. D. Stancu, Rev. Roumaine Math. Pures Appl., 15 (1970), pp. 1235-1252.

P. C. Sikkema, On some linear positive operators, Indag. Math., 32 (1970), pp. 327-337,

P. C. Sikkema, On the asymptotic approximation with operators of Meyer-König and Zeller, Indag. Math., 32 (1970), pp. 428-440,

D. D. Stancu, On the beta approximating operators of second kind, Rev. Anal. Numér. Théor. Approx., 24 (1995), pp. 231-239,

R. Upreti, Approximation properties of Beta operators, J. Approx. Theory, 45 (1985), pp. 85-89,

E. V. Voronovskaja, Détermination de la forme asymptotique de l' approximation des fonctions par les polynômes de S. Bernstein, Dokl. Akad. Nauk. SSSR, A (1932), pp. 79-85.




How to Cite

Abel, U. (1998). Asymptotic approximation with Stancu Beta operators. Rev. Anal. Numér. Théor. Approx., 27(1), 5–13. Retrieved from