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. Formally, if one is given

\sum_{n=-\infty}^\infty a_n\cdot x^n

then a multisection is a power series of the form

\sum_{m=-\infty}^\infty a_{cm%2Bd}\cdot x^{cm%2Bd}

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

Multisection of analytic functions

A multisection of the series of an analytic function

F(x) = \sum_{n=-\infty}^\infty a_n\cdot x^n

has a closed-form expression in terms of the function F(x):

\sum_{m=-\infty}^\infty a_{cm%2Bd}\cdot x^{cm%2Bd} = \tfrac{1}{c}\cdot \sum_{k=0}^{c-1} w^{-kd}\cdot F(w^k\cdot x),

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

Example

Multisection of a binomial

(1%2Bx)^q = {q\choose 0} x^0 %2B {q\choose 1} x %2B {q\choose 2} x^2 %2B \cdots

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

{q\choose d} %2B {q\choose d%2Bc} %2B {q\choose d%2B2c} %2B \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}.

References