In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence
a
n
written in the form
f
(
s
)
=
∑
n
=
0
∞
(
−
1
)
n
(
s
n
)
a
n
=
∑
n
=
0
∞
(
−
s
)
n
n
!
a
n
where
(
s
n
)
is the binomial coefficient and
(
s
)
n
is the rising factorial. Newtonian series often appear in relations of the form seen in umbral calculus.
The generalized binomial theorem gives
(
1
+
z
)
s
=
∑
n
=
0
∞
(
s
n
)
z
n
=
1
+
(
s
1
)
z
+
(
s
2
)
z
2
+
⋯
.
A proof for this identity can be obtained by showing that it satisfies the differential equation
(
1
+
z
)
d
(
1
+
z
)
s
d
z
=
s
(
1
+
z
)
s
.
The digamma function:
ψ
(
s
+
1
)
=
−
γ
−
∑
n
=
1
∞
(
−
1
)
n
n
(
s
n
)
.
The Stirling numbers of the second kind are given by the finite sum
{
n
k
}
=
1
k
!
∑
j
=
0
k
(
−
1
)
k
−
j
(
k
j
)
j
n
.
This formula is a special case of the kth forward difference of the monomial xn evaluated at x = 0:
Δ
k
x
n
=
∑
j
=
0
k
(
−
1
)
k
−
j
(
k
j
)
(
x
+
j
)
n
.
A related identity forms the basis of the Nörlund–Rice integral:
∑
k
=
0
n
(
n
k
)
(
−
1
)
k
s
−
k
=
n
!
s
(
s
−
1
)
(
s
−
2
)
⋯
(
s
−
n
)
=
Γ
(
n
+
1
)
Γ
(
s
−
n
)
Γ
(
s
+
1
)
=
B
(
n
+
1
,
s
−
n
)
where
Γ
(
x
)
is the Gamma function and
B
(
x
,
y
)
is the Beta function.
The trigonometric functions have umbral identities:
∑
n
=
0
∞
(
−
1
)
n
(
s
2
n
)
=
2
s
/
2
cos
π
s
4
and
∑
n
=
0
∞
(
−
1
)
n
(
s
2
n
+
1
)
=
2
s
/
2
sin
π
s
4
The umbral nature of these identities is a bit more clear by writing them in terms of the falling factorial
(
s
)
n
. The first few terms of the sin series are
s
−
(
s
)
3
3
!
+
(
s
)
5
5
!
−
(
s
)
7
7
!
+
⋯
which can be recognized as resembling the Taylor series for sin x, with (s)n standing in the place of xn.
In analytic number theory it is of interest to sum
∑
k
=
0
B
k
z
k
,
where B are the Bernoulli numbers. Employing the generating function its Borel sum can be evaluated as
∑
k
=
0
B
k
z
k
=
∫
0
∞
e
−
t
t
z
e
t
z
−
1
d
t
=
∑
k
=
1
z
(
k
z
+
1
)
2
.
The general relation gives the Newton series
∑
k
=
0
B
k
(
x
)
z
k
(
1
−
s
k
)
s
−
1
=
z
s
−
1
ζ
(
s
,
x
+
z
)
,
where
ζ
is the Hurwitz zeta function and
B
k
(
x
)
the Bernoulli polynomial. The series does not converge, the identity holds formally.
Another identity is
1
Γ
(
x
)
=
∑
k
=
0
∞
(
x
−
a
k
)
∑
j
=
0
k
(
−
1
)
k
−
j
Γ
(
a
+
j
)
(
k
j
)
,
which converges for
x
>
a
. This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent)
f
(
x
)
=
∑
k
=
0
(
x
−
a
h
k
)
∑
j
=
0
k
(
−
1
)
k
−
j
(
k
j
)
f
(
a
+
j
h
)
.
