Manuscript I-B-VP@3 · I-B-VP-v3.tex · proposition · 93c04eb4

Proposition: Exact quotient identities

Exact Friedrichs observables and analytical checks › Diffraction identity and quotient sums · lines 1308–1347 · proof included or linked

I-B-VP@3#prop:quotient

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For $N \geq 2$ ,

k = 1 \sum^{N - 1} csc^{2} \frac{π k}{N} = \frac{N ^{2} - 1}{3}, k = 1 \sum^{N - 1} csc^{4} \frac{π k}{N} = \frac{( N ^{2} - 1 ) ( N ^{2} + 11 )}{45} .

For $j = 0, \dots, N - 1$ , put $u = j (N - j)$ . Then

k = 1 \sum^{N - 1} e^{2 π i j k / N} csc^{2} \frac{π k}{N} = \frac{N ^{2} - 1}{3} - 2 u,

k = 1 \sum^{N - 1} e^{2 π i j k / N} csc^{4} \frac{π k}{N} = \frac{( N ^{2} - 1 ) ( N ^{2} + 11 )}{45} - \frac{2}{3} u (u + 2) .

Proof. Logarithmic differentiation of the sine product gives

k = 0 \sum^{N - 1} csc^{2} (x + \frac{π k}{N}) = N^{2} csc^{2} (N x) .

Compare constant and quadratic terms, using $\frac{1}{2} (csc^{2} x)^{''} = 3 csc^{4} x - 2 csc^{2} x$ . This proves the untwisted identities.

For the twisted sums, the discrete second difference of the first sum is $4$ away from $j = 0$ modulo $N$ , and that of the second is minus four times the first. The displayed polynomials satisfy these equations and the periodic endpoint values; the periodic homogeneous second-difference equation has only constants. The known $j = 0$ values fix those constants.

Displayed formulas in this passage

unlabeled · line 1311
$k = 1 \sum^{N - 1} csc^{2} \frac{π k}{N} = \frac{N ^{2} - 1}{3}, k = 1 \sum^{N - 1} csc^{4} \frac{π k}{N} = \frac{( N ^{2} - 1 ) ( N ^{2} + 11 )}{45} .$
unlabeled · line 1318
$k = 1 \sum^{N - 1} e^{2 π i j k / N} csc^{2} \frac{π k}{N} = \frac{N ^{2} - 1}{3} - 2 u,$
unlabeled · line 1323
$k = 1 \sum^{N - 1} e^{2 π i j k / N} csc^{4} \frac{π k}{N} = \frac{( N ^{2} - 1 ) ( N ^{2} + 11 )}{45} - \frac{2}{3} u (u + 2) .$
unlabeled · line 1333
$k = 0 \sum^{N - 1} csc^{2} (x + \frac{π k}{N}) = N^{2} csc^{2} (N x) .$

Proposition: Exact quotient identities

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