"Summing up the series a3k wherelog⁡(1−x+x2)=∑akxk" was answered by our experts

Jackson Floyd

Answered question

2022-04-02

Summing up the series

a_{3} k

where

\log (1 - x + x^{2}) = \sum a_{k} x^{k}

clarkchica44klt

Beginner2022-04-03Added 17 answers

Lemma: Let

f (x) = \sum a_{n} x^{n}

be a power series. Then

\sum a_{3 n} x^{3 n} = \frac{f (x) + f (ω x) + f (ω^{2} x)}{3}

where

ω = e^{\frac{2 π t}{3}}

Proof. Ignoring convergence it suffices to prove this for a single term, and then it boils down to the identity

\frac{1 + ω^{n} + ω^{2 n}}{3}

This is a special case of the discrete Fourier transform. Applying the lemma, we readily obtain that the desired sum is

\frac{\ln 1 + \ln (1 - ω - ω^{2}) + \ln (1 - ω^{2} + ω)}{3} = \frac{1}{3} \ln (2 ω^{2} \cdot 2 ω)

= \frac{2}{3} \ln 2

where we use the fact that

1 + ω + ω^{2} = 0

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