Asymptotic Expansions for Generalized Bernstein–Durrmeyer and Genuine Bernstein–Durrmeyer Operators

In this paper, we study an asymptotic expansion for a general family of Bernstein–Durrmeyer type operators Lₙj,i L n j , i , i,j∈ N₀ i , j ∈ N 0 . For $$i=j=1$$ i = j = 1 , these operators reduce to the genuine Bernstein–Durrmeyer operators, while for $$i=j=0$$ i = j = 0 they coincide with the classical Bernstein–Durrmeyer operators. We derive an explicit asymptotic expansion for Lₙj,i L n j , i under suitable smoothness conditions on the approximated function.