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

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

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

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.

Example

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:

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

References

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