Paper IV: Signed cyclic sections and the distinguished denominator

5 Signed cyclic sections and the distinguished denominator

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Definition 5.1

For \(0\leq j<N\), define

\[ {\mathsf D}_{N,j}(z) = \sum_{n\geq0}(-1)^n\frac{z^{Nn+j}}{(Nn+j)!}. \]

The distinguished signed denominator is

\[ {\mathsf D}_N(z)={\mathsf D}_{N,0}(z). \]

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Fix any \(\rho\in{\mathbb C}\) with \(\rho^N=-1\).

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Proposition 5.2: Relation with the transfer exponential

For every \(j\),

\[ {\mathsf D}_{N,j}(z) = \rho^{-j}{\mathsf E}_{N,j}(\rho z). \]

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Proof of Proposition 5.2

The coefficient of \(z^{Nn+j}\) on the right is

\[ \rho^{-j}\rho^{Nn+j}\frac1{(Nn+j)!} = (-1)^n\frac1{(Nn+j)!}. \]

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Theorem 5.3: Signed derivative cycle

For \(1\leq j<N\),

\[ {\mathsf D}_{N,j}'={\mathsf D}_{N,j-1}, \]

and

\[ {\mathsf D}_{N,0}'=-{\mathsf D}_{N,N-1}. \]

Every signed section satisfies

\[ {\mathsf D}_{N,j}^{(N)}=-{\mathsf D}_{N,j}. \]

The \(N\) signed sections form the basis of \(y^{(N)}=-y\) determined by

\[ {\mathsf D}_{N,j}^{(k)}(0)=\delta_{jk} \]

for \(0\leq j,k<N\).

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Proof of Theorem 5.3

The derivative formulas follow directly from the series. The first nonzero term of \({\mathsf D}_{N,j}\) is \(z^j/j!\), so the initial value matrix is the identity. The basis statement follows from the dimension of the solution space.

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Corollary 5.4: Wronskian of the signed basis

Assume \(N\geq2\). The Wronskian of

\[ {\mathsf D}_{N,0}(z),\ldots,{\mathsf D}_{N,N-1}(z) \]

is one.

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Proof of Corollary 5.4

The derivative matrix at the origin is the identity. The determinant factorization in the proof of Corollary 4.3 applies with \(a_\ell=\rho\omega^\ell\) and with the signed-section Fourier weights. Since \(\sum_\ell a_\ell=0\), the determinant is constant and equals one.

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Proposition 5.5: Twisted addition law

For \(0\leq k<N\),

\[ {\mathsf D}_{N,k}(z+w) = \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}} (-1)^{\lfloor(i+j)/N\rfloor} {\mathsf D}_{N,i}(z){\mathsf D}_{N,j}(w). \]

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Proof of Proposition 5.5

Use Proposition 5.2 and Corollary 4.4. The factor introduced by a pair \((i,j)\) is

\[ \rho^{i+j-k}. \]

Since \(0\leq i+j\leq2N-2\) and \(i+j\equiv k\bmod N\), this factor is one when \(i+j=k\) and \(-1\) when \(i+j=k+N\).

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Theorem 5.6: Finite exponential representation

Let

\[ {\mathcal R}_N=\{\eta\in{\mathbb C}\mathrel{:}\eta^N=-1\}. \]

Then

\[ {\mathsf D}_N(z) = \frac1N\sum_{\eta\in{\mathcal R}_N}e^{\eta z}. \]

More generally,

\[ {\mathsf D}_{N,j}(z) = \frac{\rho^{-j}}N \sum_{\ell=0}^{N-1} \omega^{-j\ell}e^{\rho\omega^\ell z}. \]

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Proof of Theorem 5.6

The second formula is the root filter expression for \(\rho^{-j}{\mathsf E}_{N,j}(\rho z)\). The first is the case \(j=0\).

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Definition 5.7

Define the reduced denominator

\[ \Delta_N(w) = \sum_{n\geq0}(-1)^n\frac{w^n}{(Nn)!}. \]

Then

\[ {\mathsf D}_N(z)=\Delta_N(z^N). \]

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Corollary 5.8: Symmetries

The denominator satisfies

\[ {\mathsf D}_N(\omega z)={\mathsf D}_N(z) \]

and

\[ {\mathsf D}_N(\overline z)=\overline{{\mathsf D}_N(z)}. \]

Its nonzero zeros occur in orbits under multiplication by the \(N\)th roots of unity and under complex conjugation.

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Proof of Corollary 5.8

The first identity follows from the powers \(z^{Nn}\). The second follows from the real coefficients. Both symmetries preserve the zero set and multiplicity.

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Example 5.9

The first denominators are

\[ {\mathsf D}_1(z)=e^{-z}, \qquad {\mathsf D}_2(z)=\cos z. \]

For \(N=3\),

\[ {\mathsf D}_3(z) = \frac13\left( e^{-z} +2e^{z/2}\cos\left(\frac{\sqrt3z}{2}\right) \right). \]

For \(N=4\),

\[ {\mathsf D}_4(z) = \cos\left(\frac z{\sqrt2}\right) \cosh\left(\frac z{\sqrt2}\right). \]

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