On the asymptotic approximation with bivariate operators of Bleimann, Butzer and Hahn (Q1288272)

In this interesting paper, the author considers the two-dimensional Bleiman, Butzer, Hahn operator \(L_n[f](x,y)= \sum^n_{k=0} \sum^{n-k}_{\ell=0} f({k\over n-k+1}, {\ell\over n-\ell+1}) {n\choose k,\ell} ({x\over 1+ x})^k({n \over 1+y})^\ell ({1-xy \over(1+x) (1+y)})^{n-k- \ell}\) for functions \(f\) defined on the region \(\Delta= \{(x,y)\in \mathbb{R}^2: x,y\geq 0; xy< 1\}\). The author establishes a complete asymptotic expansion \[ L_n[f] (x,y)= f(x,y) +\sum^\infty_{k=1} c_k(f;x,y) (n+1)^{-k} \] where the coefficients \(c_k (f;x,y)\) are explicitly calculated in terms of partial derivatives of \(f\) and Stirling numbers. Several interesting examples are given, some of which are of Voronovskaya type.