In mathematics, series built from equally spaced terms of another series
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
![{\displaystyle \sum _{n=-\infty }^{\infty }a_{n}\cdot z^{n))](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a65449b46f9f4720f8d1a4080c9777793b3ed4)
then its multisection is a power series of the form
![{\displaystyle \sum _{m=-\infty }^{\infty }a_{qm+p}\cdot z^{qm+p))](https://wikimedia.org/api/rest_v1/media/math/render/svg/6818e5a2eabbf641e18e9357ae71fcf10a8e3ca9)
where p, q are integers, with 0 ≤ p < q. Series multisection represents one of the common transformations of generating functions.
Multisection of analytic functions
A multisection of the series of an analytic function
![{\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}\cdot z^{n))](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9d2d5dbda876d9e29a6ce4dc74575e4b7fd883)
has a closed-form expression in terms of the function
:
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d8182b1cad8d1d07aa91ac674a40f31d550aeeb)
where
is a 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] This expression is especially useful in that it can convert an infinite sum into a finite sum. It is used, for example, in a key step of a standard proof of Gauss's digamma theorem, which gives a closed-form solution to the digamma function evaluated at rational values p/q.
Examples
Bisection
In general, the bisections of a series are the even and odd parts of the series.
Geometric series
Consider the geometric series
![{\displaystyle \sum _{n=0}^{\infty }z^{n}={\frac {1}{1-z))\quad {\text{ for ))|z|<1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/130d4294c313ac29541f984a1180b5693219b93b)
By setting
in the above series, its multisections are easily seen to be
![{\displaystyle \sum _{m=0}^{\infty }z^{qm+p}={\frac {z^{p)){1-z^{q))}\quad {\text{ for ))|z|<1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/723fcee5a659334a32bf417467bb300da80f1741)
Remembering that the sum of the multisections must equal the original series, we recover the familiar identity
![{\displaystyle \sum _{p=0}^{q-1}z^{p}={\frac {1-z^{q)){1-z)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86279bdc8036698a8d305673e3fff702fc46c423)
Exponential function
The exponential function
![{\displaystyle e^{z}=\sum _{n=0}^{\infty }{z^{n} \over n!))](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c681291c81055c8141c2e923bcb506a7e254809)
by means of the above formula for analytic functions separates into
![{\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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3082b3577956f3b301d760ed8d5be318290c3789)
The bisections are trivially the hyperbolic functions:
![{\displaystyle \sum _{m=0}^{\infty }{z^{2m} \over (2m)!}={\frac {1}{2))\left(e^{z}+e^{-z}\right)=\cosh {z))](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0d433d7c5488c3cd8acea2406e5ac29af78935)
![{\displaystyle \sum _{m=0}^{\infty }{z^{2m+1} \over (2m+1)!}={\frac {1}{2))\left(e^{z}-e^{-z}\right)=\sinh {z}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/749e4bbb0976dd5de3595486de929cee9d216155)
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
![{\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)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3929fde905d64781775eacf19b12ea1c5a27a91)
These can be seen as solutions to the linear differential equation
with boundary conditions
, using Kronecker delta notation. In particular, the trisections are
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6201d31d18856a621e661c2d3b7cc074868b1b6b)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/272b961cfb1bd42f40f29ab98c1ab2391cebb878)
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a10e9193dfb07a8d80e580368c0310857ff344c)
and the quadrisections are
![{\displaystyle \sum _{m=0}^{\infty }{z^{4m} \over (4m)!}={\frac {1}{2))\left(\cosh {z}+\cos {z}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa00b6c9967688a4e3cda484d94c78b52a5f2f8)
![{\displaystyle \sum _{m=0}^{\infty }{z^{4m+1} \over (4m+1)!}={\frac {1}{2))\left(\sinh {z}+\sin {z}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e69274ffd3b6534d819abfeb8513dce9740d8de1)
![{\displaystyle \sum _{m=0}^{\infty }{z^{4m+2} \over (4m+2)!}={\frac {1}{2))\left(\cosh {z}-\cos {z}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f33161a9a1a7a0784b00a42951da8baac0200568)
![{\displaystyle \sum _{m=0}^{\infty }{z^{4m+3} \over (4m+3)!}={\frac {1}{2))\left(\sinh {z}-\sin {z}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c678db0f1507d151478740eeb38c61bf865b8a67)
Binomial series
Multisection of a binomial expansion
![{\displaystyle (1+x)^{n}={n \choose 0}x^{0}+{n \choose 1}x+{n \choose 2}x^{2}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/95f5d59812b2aac9ebf494d8e53847f2dd6c0e29)
at x = 1 gives the following identity for the sum of binomial coefficients with step 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)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e55d08539f30dd6552c57deba969d246b5bd769)