Some Bernstein-Durrmeyer-type operators (Q1383103)

With a view to generalize Bernstein and Szász operators \textit{A. Meir} and \textit{A. Sharma} [Indag. Math. 29, 395-403 (1967; Zbl 0176.34801)] had introduced two linear positive operators, the first one being based on Laguerre polynomials while the second on Hermite polynomials. In the present paper the authors define three Bernstein-Durrmeyer-type operators based on the above operators. Their first operator \(M_n^{(\lambda, \alpha)}\) is defined as: \[ M_n^{(\lambda, \alpha)} (f;x)= {n+1 \over L_n^{(\alpha)} (\lambda)} \sum^n_{k=0} {n+\alpha \choose n-k} L_k^{(\alpha)} \left({\lambda \over x} \right) x^k(1-x)^{n-k} \int^1_0 p_{n,k} (t)f(t)dt, \] where \(p_{n,k} (t)= {n\choose k} t^k(1-t)\) and \(L_n^{(\alpha)} (\lambda)\) is the Laguerre polynomial of degree \(n\). They denote their other two operators by \(P_n^{(\lambda, \alpha)}\) and \(\widetilde S_n^\lambda\). For each operator they establish a Voronoskaja-type theorem.