The most important coordinate change is \(w=z^N\). A term of degree \(n\) in \(w\) has degree \(Nn\) in \(z\). Its coefficient does not change, but an exponential-generating-function label does.
\[
\underbrace{b_{N,n}(r)\,w^n}_{\text{ordinary coefficient}}
=\underbrace{P_{N,n}(r)\frac{z^{Nn}}{(Nn)!}}_{\text{block exponential coefficient}},
\qquad P_{N,n}(r)=(Nn)!b_{N,n}(r).
\]
| Object | Definition and role |
|---|---|
| \(E_{N,j}(z)\) | \(\displaystyle\sum_{n\ge0}\frac{z^{Nn+j}}{(Nn+j)!}\) selects one residue class of the exponential. |
| \(D_{N,j}(z)\) | The same series with \((-1)^n\). One full derivative cycle contributes a minus sign. |
| \(D_N(z)\) | \(D_{N,0}(z)\), the denominator whose zeros control the signed reciprocal family. |
| \(\Delta_N(w)\) | \(\displaystyle\sum_{n\ge0}\frac{(-1)^nw^n}{(Nn)!}\). Thus \(D_N(z)=\Delta_N(z^N)\). |
| \(P_{N,n}(r)\) | Block-EGF coefficients of \(D_N^{-r}\). They are polynomials in \(r\). |
| \(\kappa_{N,m}\) | \(\displaystyle-\log\Delta_N(w)=\sum_{m\ge1}\kappa_{N,m}w^m/m\). |
| \(T_N(z)\) | \(-D_N'(z)/D_N(z)\), a function, not the coefficient \(T_N(n)\). |
| \(\Phi_{N,j}(w)\) | \(\displaystyle j!\sum_{n\ge0}\frac{w^n}{(Nn+j)!}\) removes the forced zero of \(E_{N,j}\). |
| \(C_{N,j}(z)\) | \(1/\Phi_{N,j}(z^N)\). This is generally not \(1/D_N(z)\). |
| \(\Lambda_N\) | Zeros of \(\Delta_N\), counted with multiplicity, one per rotational orbit of zeros of \(D_N\). |
A normalization test
The coefficient \(5/24\) of \(w^2\) in \(\Delta_2(w)^{-1}\) is the same term as \(5z^4/4!\). It is not \(5w^2/2!\). Keeping the two factorials separate prevents errors in Bell formulas, convolution weights, and zero-moment reconstruction.
Read this as
Use \(z\) for differentiation and rotational geometry. Use \(w\) for reciprocal recurrences, canonical products, and orbit-zero sums.