In the last post, we came to the conclusion that E=∑n−1j=0(j+n2j+1)E(j)0. This is an n-1'th degree inhomogenous linear equation. To solve it, we need to solve the characteristic equation ∑n−1j=0(j+n2j+1)rj=0 for r.
It seems that, in general, the solutions to the characteristic equation are ri=−4cos2πi2n(1≤i<n) but I don't have a proof for this yet. I found the solutions by chance: by noticing that the solutions for low n involved the same square roots as cos did.
As I don't have a proof yet, I'll give you an example instead. Let n=4. Then the characteristic equation is
0=3∑j=0(j+42j+1)rj=(41)+(53)r+(65)r2+(77)r3=4+10r+6r2+r3=(2+r)(2+4r+r2)=(2+r)(2+√2+r)(2−√2+r)
whence we may read off the solutions r=−2,−2−√2,−2+√2. This is just what the formula predicts:
r1=−4cos2π8=−412(1+1√2)=−2−√2r2=−4cos2π4=−412=−2r3=−4cos23π8=−412(1−1√2)=−2+√2
In conclusion, the formula is reasonable and will do without a proof for the moment.
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