# Series multisection

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

${\displaystyle \sum _{n=-\infty }^{\infty }a_{n}\cdot x^{n}}$

then its multisection is a power series of the form

${\displaystyle \sum _{m=-\infty }^{\infty }a_{cm+d}\cdot x^{cm+d}}$

where c, d are integers, with 0 ≤ d < c.

## Multisection of analytic functions

A multisection of the series of an analytic function

${\displaystyle F(x)=\sum _{n=-\infty }^{\infty }a_{n}\cdot x^{n}}$

has a closed-form expression in terms of the function ${\displaystyle F(x)}$:

${\displaystyle \sum _{m=-\infty }^{\infty }a_{cm+d}\cdot x^{cm+d}={\tfrac {1}{c}}\cdot \sum _{k=0}^{c-1}w^{-kd}\cdot F(w^{k}\cdot x),}$

where ${\displaystyle w=e^{\frac {2\pi i}{c}}}$ is a primitive c-th root of unity.

### Example

Multisection of a binomial expansion

${\displaystyle (1+x)^{q}={q \choose 0}x^{0}+{q \choose 1}x+{q \choose 2}x^{2}+\cdots }$

at x = 1 gives the following identity for the sum of binomial coefficients with step c:

${\displaystyle {q \choose d}+{q \choose d+c}+{q \choose d+2c}+\cdots ={\frac {1}{c}}\cdot \sum _{k=0}^{c-1}\left(2\cos {\frac {\pi k}{c}}\right)^{q}\cdot \cos {\frac {\pi (q-2d)k}{c}}.}$