Series multisection

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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

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

then its multisection is a power series of the form

\sum_{m=-\infty}^\infty a_{cm+d}\cdot x^{cm+d}

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

Multisection of analytic functions[edit]

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+d}\cdot x^{cm+d} = \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.


Multisection of a binomial

(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:

{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}.