AbstractBinomial operators are the most important extension to Bernstein operators, defined by $$ \bigl(L^{Q}_{n} f\bigr) (x)=\frac{1}{b_{n}(1)} \sum ^{n}_{k=0}\binom { n}{k } b_{k}(x)b_{n-k}(1-x)f\biggl( \frac{k}{n}\biggr),\quad f\in C[0, 1], $$
(
L
n
Q
f
)
(
x
)
=
1
b
n
(
1
)
∑
k
=
0
n
(
n
k
)
b
k
(
x
)
b
n
−
k
(
1
−
x
)
f
(
k
n
)
,
f
∈
C
[
0
,
1
]
,
where $\{b_{n}\}$
{
b
n
}
is a sequence of binomial polynomials associated to a delta operator Q. In this paper, we discuss the binomial operators $\{L^{Q}_{n} f\}$
{
L
n
Q
f
}
preservation such as smoothness and semi-additivity by the aid of binary representation of the operators, and present several illustrative examples. The results obtained in this paper generalize what are known as the corresponding Bernstein operators.