In mathematics, amultisection of a power series is a newpower series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series

${\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_n\cdot z^n}$

then its multisection is a power series of the form

${\displaystyle \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_{qm+p}\cdot z^{qm+p}}$

wherep,q are integers, with 0 ≤p <q. Series multisection represents one of the commontransformations of generating functions.

A multisection of the series of ananalytic function

${\displaystyle f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n\cdot z^n}$

has aclosed-form expression in terms of the function${\displaystyle f(x)}$:

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}{a}_{qm+p}\cdot z^{qm+p}=\frac{1}{q}\cdot \sum _{k=0}^{q-1}{\omega }^{-kp}\cdot f({\omega }^k\cdot z),}$

where${\displaystyle \omega =e^{\frac{2\pi i}{q}}}$ is aprimitiveq-th root of unity. This expression is often called a root of unity filter. This solution was first discovered byThomas Simpson.^[1]

In general, the bisections of a series are theeven and odd parts of the series.

Consider thegeometric series

By setting${\displaystyle z\to z^q}$ in the above series, its multisections are easily seen to be

Remembering that the sum of the multisections must equal the original series, we recover the familiar identity

${\displaystyle \sum _{p=0}^{q-1}{z}^p=\frac{1-z^q}{1-z}.}$

The exponential function

${\displaystyle e^z=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}}$

by means of the above formula for analytic functions separates into

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}\frac{z^{qm+p}}{(qm+p)!}=\frac{1}{q}\cdot \sum _{k=0}^{q-1}{\omega }^{-kp}{e}^{{\omega }^{k}z}.}$

The bisections are trivially thehyperbolic functions:

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}\frac{z^{2m}}{(2m)!}=\frac{1}{2}\left(e^z+e^{-z}\right)=\cosh z}$

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}\frac{z^{2m+1}}{(2m+1)!}=\frac{1}{2}\left(e^z-e^{-z}\right)=\sinh z.}$

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

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}\frac{z^{qm+p}}{(qm+p)!}=\frac{1}{q}\cdot \sum _{k=0}^{q-1}{e}^{z\cos (2\pi k/q)}\cos \left(z\sin \left(\frac{2\pi k}{q}\right)-\frac{2\pi kp}{q}\right).}$

These can be seen as solutions to thelinear differential equation${\displaystyle f^{(q)}(z)=f(z)}$ withboundary conditions${\displaystyle f^{(k)}(0)={\delta }_{k,p}}$, usingKronecker delta notation. In particular, the trisections are

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}\frac{z^{3m}}{(3m)!}=\frac{1}{3}\left(e^z+2e^{-z/2}\cos \frac{\sqrt{3}z}{2}\right)}$

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}\frac{z^{3m+1}}{(3m+1)!}=\frac{1}{3}\left(e^z-2e^{-z/2}\cos \left(\frac{\sqrt{3}z}{2}+\frac{\pi }{3}\right)\right)}$

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}\frac{z^{3m+2}}{(3m+2)!}=\frac{1}{3}\left(e^z-2e^{-z/2}\cos \left(\frac{\sqrt{3}z}{2}-\frac{\pi }{3}\right)\right),}$

and the quadrisections are

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}\frac{z^{4m}}{(4m)!}=\frac{1}{2}\left(\cosh z+\cos z\right)}$

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}\frac{z^{4m+1}}{(4m+1)!}=\frac{1}{2}\left(\sinh z+\sin z\right)}$

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}\frac{z^{4m+2}}{(4m+2)!}=\frac{1}{2}\left(\cosh z-\cos z\right)}$

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}\frac{z^{4m+3}}{(4m+3)!}=\frac{1}{2}\left(\sinh z-\sin z\right).}$

Multisection of abinomial expansion

${\displaystyle (1+x{)}^n=(\genfrac{}{}{0ex}{}{n}{0})x^0+(\genfrac{}{}{0ex}{}{n}{1})x+(\genfrac{}{}{0ex}{}{n}{2})x^2+\cdots }$

atx = 1 gives the following identity for the sum ofbinomial coefficients with stepq:

${\displaystyle (\genfrac{}{}{0ex}{}{n}{p})+(\genfrac{}{}{0ex}{}{n}{p+q})+(\genfrac{}{}{0ex}{}{n}{p+2q})+\cdots =\frac{1}{q}\cdot \sum _{k=0}^{q-1}{\left(2\cos \frac{\pi k}{q}\right)}^n\cdot \cos \frac{\pi (n-2p)k}{q}.}$

Series multisection converts an infinite sum into a finite sum. It is used, for example, in a key step of a standard proof ofGauss's digamma theorem, which gives a closed-form solution to the digamma function evaluated at rational valuesp/q.

• Weisstein, Eric W."Series Multisection".MathWorld.
• Somos, MichaelA Multisection of q-Series, 2006.
• John Riordan (1968).Combinatorial identities. New York: John Wiley and Sons.

• Algebra
• Combinatorics
• Mathematical analysis
• Complex analysis
• Series (mathematics)

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