Summing up the series \(\displaystyle{a}_{{3}}{k}\) where \(\displaystyle{\log{{\left({1}-{x}+{x}^{{2}}\right)}}}=\sum{a}_{{k}}{x}^{{k}}\)

Jackson Floyd

Jackson Floyd

Answered question

2022-04-02

Summing up the series a3k where
log(1x+x2)=akxk

Answer & Explanation

clarkchica44klt

clarkchica44klt

Beginner2022-04-03Added 17 answers

Lemma: Let f(x)=anxn be a power series. Then
a3nx3n=f(x)+f(ωx)+f(ω2x)3
where ω=e2πt3
Proof. Ignoring convergence it suffices to prove this for a single term, and then it boils down to the identity
1+ωn+ω2n3
This is a special case of the discrete Fourier transform. Applying the lemma, we readily obtain that the desired sum is
ln1+ln(1ωω2)+ln(1ω2+ω)3=13ln(2ω22ω)
=23ln2
where we use the fact that 1+ω+ω2=0

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