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While solving a programming problem, I stumbled across this sum.

$\sum_{i=0}^{\lfloor\frac{n}{k}\rfloor}{n \choose i * k}$

I know this can be solved by taking the $k^{th}$ root of unity and then summing them up so that only the multiples of $k$ remains. But what I can not figure out is how to proceed further and get a closed formula so that I can code it.

Thanks in advance.

Martin Sleziak
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smashingpumpkin
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1 Answers1

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Let $\omega$ be a primitive $k$th root of unity, so that for any integer $b$,

$$\frac1k\sum_{j=1}^k \omega^{bj} = \begin{cases}1,&k\mid b, \\ 0,&k\nmid b\end{cases}.$$

This allows us to write $\sum_{i=0}^{\lfloor\frac{n}{k}\rfloor}{n \choose i * k}$ as $$ \sum_{\substack{0\le b\le n,\\b \equiv 0 \pmod k}}{n \choose b} = \sum_{b=0}^n {n \choose b} \frac1k\sum_{j=1}^k \omega^{bj} = \frac1k \sum_{j=1}^k \sum_{b=0}^n {n\choose b} \omega^{bj}.$$

The innermost sum can be recognized as just the binomial expansion for $(1+\omega^j)^n$. So that means the desired sum is simply

$$\frac1k \sum_{j=1}^k (1+\omega^j)^n.$$

Now if you're coding this you might want to put some thought into what number system to evaluate $\omega$ in. You could do it using floating-point complex arithmetic but if $n$ is large you'll get a lot of precision loss in evaluating $(1+\omega^j)^n$ (but at least the end result can be rounded to an integer, albeit one with many digits). You could alternatively evaluate over different finite fields (say $\mathbb F_p$ where $p \equiv 1 \pmod k$ so that admits $k$th roots of unity), and then Chinese remainder the moduli together.

Erick Wong
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  • Can you please elaborate on " alternatively evaluate over different finite fields" and the field you mentioned or provide a link where I can read up more about this topic? Thank you. @ErickWong – smashingpumpkin Aug 25 '16 at 09:53
  • @smashingpumpkin The value you're looking for is an integer, so one could in principle ask 'what is the value mod $p$?" for a given prime $p$ For certain primes $p$ it is possible to interpret $\omega$ as an integer instead of a complex number. Then you can easily calculate the expression mod $p$. Do this for enough different $p$ and you can extract the original value by combining the results (you need an upper bound on the size of the number). – Erick Wong Aug 25 '16 at 14:59
  • A hybrid approach would be to use complex-valued arithmetic to get an approximation, determine an error interval for the estimate, then you only need enough primes to cover the length of the interval. – Erick Wong Aug 25 '16 at 16:24