Successive remainders of the Newton series
Authors: Crofts G. W. , Shaw J. K. 
Published: July 2010

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals f left-parenthesis lamda 1 right-parenthesis plus sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts normal upper Delta Subscript lamda Superscript n Baseline f left-parenthesis lamda Subscript n plus 1 Baseline right-parenthesis left-parenthesis z minus lamda 1 right-parenthesis left-parenthesis z minus lamda 2 right-parenthesis midline-horizontal-ellipsis left-parenthesis z minus lamda Subscript n Baseline right-parenthesis">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript lamda Superscript k Baseline f left-parenthesis z right-parenthesis equals normal upper Delta Subscript lamda Superscript k Baseline f left-parenthesis lamda Subscript k plus 1 Baseline right-parenthesis plus sigma-summation Underscript n equals k plus 1 Overscript normal infinity Endscripts normal upper Delta Subscript lamda Superscript n Baseline f left-parenthesis lamda Subscript n plus 1 Baseline right-parenthesis left-parenthesis z minus lamda Subscript k plus 1 Baseline right-parenthesis midline-horizontal-ellipsis left-parenthesis z minus lamda Subscript n Baseline right-parenthesis">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts normal upper Delta Subscript lamda Superscript n Baseline f left-parenthesis z Subscript n Baseline right-parenthesis upper C Subscript n Baseline left-parenthesis z right-parenthesis">

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}}

.