If
f
f
is analytic in the open unit disc
D
D
and
λ
\lambda
is a sequence of points in
D
D
converging to 0, then
f
f
admits the Newton series expansion
f
(
z
)
=
f
(
λ
1
)
+
∑
n
=
1
∞
Δ
λ
n
f
(
λ
n
+
1
)
(
z
−
λ
1
)
(
z
−
λ
2
)
⋯
(
z
−
λ
n
)
f(z) = f({\lambda _1}) + \sum \nolimits _{n = 1}^\infty {\Delta _\lambda ^nf({\lambda _{n + 1}})(z - {\lambda _1})(z - {\lambda _2}) \cdots (z - {\lambda _n})}
, where
Δ
λ
n
f
(
z
)
\Delta _\lambda ^nf(z)
is the
n
n
th divided difference of
f
f
with respect to the sequence
λ
\lambda
. The Newton series reduces to the Maclaurin series in case
λ
n
≡
0
{\lambda _n} \equiv 0
. The present paper investigates relationships between the behavior of zeros of the normalized remainders
Δ
λ
k
f
(
z
)
=
Δ
λ
k
f
(
λ
k
+
1
)
+
∑
n
=
k
+
1
∞
Δ
λ
n
f
(
λ
n
+
1
)
(
z
−
λ
k
+
1
)
⋯
(
z
−
λ
n
)
\Delta _\lambda ^kf(z) = \Delta _\lambda ^kf({\lambda _{k + 1}}) + \sum \nolimits _{n = k + 1}^\infty {\Delta _\lambda ^nf({\lambda _{n + 1}})(z - {\lambda _{k + 1}}) \cdots (z - {\lambda _n})}
of the Newton series and zeros of the normalized remainders
∑
n
=
k
∞
a
n
z
n
−
k
\sum \nolimits _{n = k}^\infty {{a_n}{z^{n - k}}}
of the Maclaurin series for
f
f
. Let
C
λ
{C_\lambda }
be the supremum of numbers
c
>
0
c > 0
such that if
f
f
is analytic in
D
D
and each of
Δ
λ
k
f
(
z
)
,
0
⩽
k
>
∞
\Delta _\lambda ^kf(z),\;0 \leqslant k > \infty
, has a zero in
|
z
|
⩽
c
|z| \leqslant c
, then
f
≡
0
f \equiv 0
. The corresponding constant for the Maclaurin series (
C
λ
{C_\lambda }
, where
λ
n
≡
0
{\lambda _n} \equiv 0
) is called the Whittaker constant for remainders and is denoted by
W
W
. We prove that
C
λ
⩾
W
{C_\lambda } \geqslant W
, for all
λ
\lambda
, and, moreover,
C
λ
=
W
{C_\lambda } = W
if
λ
∈
l
1
\lambda \in {l_1}
. In obtaining this result, we prove that functions
f
f
analytic in
D
D
have expansions of the form
f
(
z
)
=
∑
n
=
0
∞
Δ
λ
n
f
(
z
n
)
C
n
(
z
)
f(z) = \sum \nolimits _{n = 0}^\infty {\Delta _\lambda ^nf({z_n}){C_n}(z)}
, where
|
z
n
|
⩽
W
|{z_n}| \leqslant W
, for all
n
n
, and
C
n
(
z
)
{C_n}(z)
is a polynomial of degree
n
n
determined by the conditions
Δ
λ
j
C
k
(
z
j
)
=
δ
j
k
\Delta _\lambda ^j{C_k}({z_j}) = {\delta _{jk}}
.