The complete asymptotic expansion for the Meyer-König and Zeller operators (Q1359605)

While the complete asymptotic expansion of the Bernstein polynomials already appeared in a paper of Bernstein in 1932, there was no progress in this direction for the closely related operators \(M_n\) of Meyer-König and Zeller. Therefore, the purpose of the recent paper is to give the complete asymptotic expansion in the form \(M_n(f(t);x)\sim f(x)+\sum^\infty_{k=1}a_k(f;x)n^{-k}\) as \(n\) tends to infinity, provided \(f\) possesses derivatives of sufficiently high order at \(x\in[0,1]\). All coefficients \(a_k(f;x)\) are calculated explicitly in terms of \(f^{(p)}(x)\) \((p=1,2,\dots,2k)\) and in terms of Stirling numbers of the first and second kind.