Multipole radiation

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

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

then its multisection is a power series of the form

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

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

Multisection of analytic functions

A multisection of the series of an analytic function

$F(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_n\cdot x^n$

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

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

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

Example

Multisection of a binomial

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

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

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

References

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• Somos, M. A Multisection of q-Series, 2006.
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