In mathematics, a multisection of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series
∑
n
=
−
∞
∞
a
n
⋅
z
n
{\displaystyle \sum _{n=-\infty }^{\infty }a_{n}\cdot z^{n}}
then its multisection is a power series of the form
∑
m
=
−
∞
∞
a
q
m
+
p
⋅
z
q
m
+
p
{\displaystyle \sum _{m=-\infty }^{\infty }a_{qm+p}\cdot z^{qm+p}}
where p , q are integers, with 0 ≤ p < q . Series multisection represents one of the common transformations of generating functions .
Multisection of analytic functions
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A multisection of the series of an analytic function
f
(
z
)
=
∑
n
=
0
∞
a
n
⋅
z
n
{\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}\cdot z^{n}}
has a closed-form expression in terms of the function
f
(
x
)
{\displaystyle f(x)}
∑
m
=
0
∞
a
q
m
+
p
⋅
z
q
m
+
p
=
1
q
⋅
∑
k
=
0
q
−
1
ω
−
k
p
⋅
f
(
ω
k
⋅
z
)
,
{\displaystyle \sum _{m=0}^{\infty }a_{qm+p}\cdot z^{qm+p}={\frac {1}{q}}\cdot \sum _{k=0}^{q-1}\omega ^{-kp}\cdot f(\omega ^{k}\cdot z),}
where
ω
=
e
2
π
i
q
{\displaystyle \omega =e^{\frac {2\pi i}{q}}}
primitive q -th root of unity . This expression is often called a root of unity filter. This solution was first discovered by Thomas Simpson .[1] Gauss's digamma theorem , which gives a closed-form solution to the digamma function evaluated at rational values p /q .
Examples
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Bisection
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In general, the bisections of a series are the even and odd parts of the series.
Geometric series
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Consider the geometric series
∑
n
=
0
∞
z
n
=
1
1
−
z
for
|
z
|
<
1.
{\displaystyle \sum _{n=0}^{\infty }z^{n}={\frac {1}{1-z}}\quad {\text{ for }}|z|<1.}
By setting
z
→
z
q
{\displaystyle z\rightarrow z^{q}}
∑
m
=
0
∞
z
q
m
+
p
=
z
p
1
−
z
q
for
|
z
|
<
1.
{\displaystyle \sum _{m=0}^{\infty }z^{qm+p}={\frac {z^{p}}{1-z^{q}}}\quad {\text{ for }}|z|<1.}
Remembering that the sum of the multisections must equal the original series, we recover the familiar identity
∑
p
=
0
q
−
1
z
p
=
1
−
z
q
1
−
z
.
{\displaystyle \sum _{p=0}^{q-1}z^{p}={\frac {1-z^{q}}{1-z}}.}
Exponential function
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The exponential function
e
z
=
∑
n
=
0
∞
z
n
n
!
{\displaystyle e^{z}=\sum _{n=0}^{\infty }{z^{n} \over n!}}
by means of the above formula for analytic functions separates into
∑
m
=
0
∞
z
q
m
+
p
(
q
m
+
p
)
!
=
1
q
⋅
∑
k
=
0
q
−
1
ω
−
k
p
e
ω
k
z
.
{\displaystyle \sum _{m=0}^{\infty }{z^{qm+p} \over (qm+p)!}={\frac {1}{q}}\cdot \sum _{k=0}^{q-1}\omega ^{-kp}e^{\omega ^{k}z}.}
The bisections are trivially the hyperbolic functions :
∑
m
=
0
∞
z
2
m
(
2
m
)
!
=
1
2
(
e
z
+
e
−
z
)
=
cosh
z
{\displaystyle \sum _{m=0}^{\infty }{z^{2m} \over (2m)!}={\frac {1}{2}}\left(e^{z}+e^{-z}\right)=\cosh {z}}
∑
m
=
0
∞
z
2
m
+
1
(
2
m
+
1
)
!
=
1
2
(
e
z
−
e
−
z
)
=
sinh
z
.
{\displaystyle \sum _{m=0}^{\infty }{z^{2m+1} \over (2m+1)!}={\frac {1}{2}}\left(e^{z}-e^{-z}\right)=\sinh {z}.}
Higher order multisections are found by noting that all such series must be real-valued along the real line. By taking the real part and using standard trigonometric identities, the formulas may be written in explicitly real form as
∑
m
=
0
∞
z
q
m
+
p
(
q
m
+
p
)
!
=
1
q
⋅
∑
k
=
0
q
−
1
e
z
cos
(
2
π
k
/
q
)
cos
(
z
sin
(
2
π
k
q
)
−
2
π
k
p
q
)
.
{\displaystyle \sum _{m=0}^{\infty }{z^{qm+p} \over (qm+p)!}={\frac {1}{q}}\cdot \sum _{k=0}^{q-1}e^{z\cos(2\pi k/q)}\cos {\left(z\sin {\left({\frac {2\pi k}{q}}\right)}-{\frac {2\pi kp}{q}}\right)}.}
These can be seen as solutions to the linear differential equation
f
(
q
)
(
z
)
=
f
(
z
)
{\displaystyle f^{(q)}(z)=f(z)}
boundary conditions
f
(
k
)
(
0
)
=
δ
k
,
p
{\displaystyle f^{(k)}(0)=\delta _{k,p}}
Kronecker delta notation. In particular, the trisections are
∑
m
=
0
∞
z
3
m
(
3
m
)
!
=
1
3
(
e
z
+
2
e
−
z
/
2
cos
3
z
2
)
{\displaystyle \sum _{m=0}^{\infty }{z^{3m} \over (3m)!}={\frac {1}{3}}\left(e^{z}+2e^{-z/2}\cos {\frac {{\sqrt {3}}z}{2}}\right)}
∑
m
=
0
∞
z
3
m
+
1
(
3
m
+
1
)
!
=
1
3
(
e
z
−
e
−
z
/
2
(
cos
3
z
2
−
3
sin
3
z
2
)
)
{\displaystyle \sum _{m=0}^{\infty }{z^{3m+1} \over (3m+1)!}={\frac {1}{3}}\left(e^{z}-e^{-z/2}\left(\cos {\frac {{\sqrt {3}}z}{2}}-{\sqrt {3}}\sin {\frac {{\sqrt {3}}z}{2}}\right)\right)}
∑
m
=
0
∞
z
3
m
+
2
(
3
m
+
2
)
!
=
1
3
(
e
z
−
e
−
z
/
2
(
cos
3
z
2
+
3
sin
3
z
2
)
)
,
{\displaystyle \sum _{m=0}^{\infty }{z^{3m+2} \over (3m+2)!}={\frac {1}{3}}\left(e^{z}-e^{-z/2}\left(\cos {\frac {{\sqrt {3}}z}{2}}+{\sqrt {3}}\sin {\frac {{\sqrt {3}}z}{2}}\right)\right),}
and the quadrisections are
∑
m
=
0
∞
z
4
m
(
4
m
)
!
=
1
2
(
cosh
z
+
cos
z
)
{\displaystyle \sum _{m=0}^{\infty }{z^{4m} \over (4m)!}={\frac {1}{2}}\left(\cosh {z}+\cos {z}\right)}
∑
m
=
0
∞
z
4
m
+
1
(
4
m
+
1
)
!
=
1
2
(
sinh
z
+
sin
z
)
{\displaystyle \sum _{m=0}^{\infty }{z^{4m+1} \over (4m+1)!}={\frac {1}{2}}\left(\sinh {z}+\sin {z}\right)}
∑
m
=
0
∞
z
4
m
+
2
(
4
m
+
2
)
!
=
1
2
(
cosh
z
−
cos
z
)
{\displaystyle \sum _{m=0}^{\infty }{z^{4m+2} \over (4m+2)!}={\frac {1}{2}}\left(\cosh {z}-\cos {z}\right)}
∑
m
=
0
∞
z
4
m
+
3
(
4
m
+
3
)
!
=
1
2
(
sinh
z
−
sin
z
)
.
{\displaystyle \sum _{m=0}^{\infty }{z^{4m+3} \over (4m+3)!}={\frac {1}{2}}\left(\sinh {z}-\sin {z}\right).}
Binomial series
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Multisection of a binomial expansion
(
1
+
x
)
n
=
(
n
0
)
x
0
+
(
n
1
)
x
+
(
n
2
)
x
2
+
⋯
{\displaystyle (1+x)^{n}={n \choose 0}x^{0}+{n \choose 1}x+{n \choose 2}x^{2}+\cdots }
at x = 1 gives the following identity for the sum of binomial coefficients with step q :
(
n
p
)
+
(
n
p
+
q
)
+
(
n
p
+
2
q
)
+
⋯
=
1
q
⋅
∑
k
=
0
q
−
1
(
2
cos
π
k
q
)
n
⋅
cos
π
(
n
−
2
p
)
k
q
.
{\displaystyle {n \choose p}+{n \choose p+q}+{n \choose p+2q}+\cdots ={\frac {1}{q}}\cdot \sum _{k=0}^{q-1}\left(2\cos {\frac {\pi k}{q}}\right)^{n}\cdot \cos {\frac {\pi (n-2p)k}{q}}.}
References
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