Evaluating A series in a closed form

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The series is similar to that of exp(x), except for the signs which have a period of 6 in the series. We can fix this by considering the functions

f_k(x) = exp(s_k x)

where the s_k are the 6th roots of unity:

s_k = exp(pi i k/3)

for k in {0,1,2,3,4,5}.

To compute the series, we need to find that linear combination of the fk(x) that yields to correct series. This amounts to solving the following discrete Fourier problem. On the set {0,1,2,3,4,5} we have a function g(n) = 1 for 0<=n<=2 and g(n) = -1 for 3<=n<=5. We want to decompose g(n) as:

g(n) = sum from k = 0 to 5 of c_k s_k(n)

where the functions s_k(n) are defined as:

s_k(n) = s_k^(n) = exp(pi i k n/3)

To ...