# Solution 86: Double Factorial and Roots of Unity Filter (Proof)

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**Published on Apr 4, 2019**- We prove the roots of unity filter and apply it to evaluate a fascinating summation.

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Jeffrey HershMonth agoThis inspired me to try to find the general solution for any S(a) = Sum[1/(2an)!!, {n,0,Infinity}] where a is a positive integer.

Turns out with some cleaver manipulation of the sum after applying the filter it generalizes to:

S(a) = (1/a) Sum[e^((1/2) cos (k theta)) e^((i/2) sin (k theta)),{k,0,a-1}] = (1/2) Sum[f(k),{k,0,a-1}] (where theta is 2 pi/a)

which simplifies by noticing that f(k) + f(a-k) = 2 e^((1/2) cos(k theta)) * cos[(1/2) sin (k theta)]

thus (after manipulating and shifting the index of the sum)

S(a) = (1/a)(e^1/2) + (2/a) Sum[e^((1/2) cos(m theta)) * cos((1/2) sin(m theta)),{m,1,k}]

where if a is even, k = (a/2), and if a is odd k = (1/2)(a-1)

and theta = 2 pi/a

Plugging in a = 3 (as in the video) we get

S(3) = (1/3)(e^(1/2)) + (2/3) e^(-1/4) cos(sqrt(3)/4)

Hilda GaniMonth ago^{+1}Can you make a video about Roots of Unity filter?

Hiletso Eu2 months agoExcellent video

Sitanshu Chaudhary2 months ago^{+2}Make a only video on roots of unity filter please

Jeff Ahn2 months ago^{+1}Roots of unity filter -- just amazing!

Aaron He2 months agoSurprising how a square root ended up inside the cosine function!

Richard Reynolds2 months ago^{+2}An unusual radian angle answer; but if it works, it works.

nightrider2 months ago^{+1}It is not unusual. It is a special case of the discrete Fourier expansion.

steamroller822 months agoBeautiful. Well done sir!

Johannes H2 months ago^{+2}Awesome, but I couldn't gome up with that myself

Davide2 months ago^{+2}Neat

Risu0chan2 months ago^{+12}I love it so much when I learn new techniques. And it comes with a well explained proof

Chan Dan2 months ago^{+8}Can you refer any book where I can find cube root of unity filter. I have never heard of it !

Chan Dan2 months ago@nightrider sorry sir ! I have not taken the linear algebra course.

nightrider2 months ago^{+1}@Chan Dan Do you see that our problem is but the orthonormality between the zero'th basis vector and all the other basis vectors (unit roots) of the discrete Fourier transform?

nightrider2 months ago^{+1}@Chan Dan Just check the Wikipedia page for the discrete Fourier transform, particularly the orthogonality of the basis en.wikipedia.org/wiki/Discrete_Fourier_transform#Orthogonality and the references therein. That is a special case of the expansion of the Dirac or Kronecker delta function as I mentioned before.

Chan Dan2 months ago@nightrider ok ! Can you provide any specifications like author or publisher etc. ? I did not get what fourier analysis to do with cube roots of unity.

nightrider2 months ago^{+1}You can check out discrete or fast Fourier analysis. Any summation expansion of the Dirac delta function is a generalization.