2 A map of the construction
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The central idea
The same object is read in different coordinates. Selecting residues makes its Taylor coefficients visible. A signed shift makes the differential system finite. Passing to \(w=z^N\) removes rotational repetition from its zeros. The formulas below identify these descriptions under their stated hypotheses.
Two variables appear throughout the paper. The variable \(z\) carries the differential equation and the rotational symmetry. The reduced variable
records one representative from each rotational orbit. Confusing these variables hides the main simplification. The denominator \({\mathsf D}_N\) has order one as a function of \(z\), while \(\Delta_N\) has order \(1/N\) as a function of \(w\).
The analytic objects are introduced in the following order. First,
is the residue \(j\) part of the exponential. Second,
is the signed basis for \(y^{(N)}=-y\). The section \({\mathsf D}_N={\mathsf D}_{N,0}\) is the denominator used in the reciprocal family
Set \({\mathsf T}_N=-{\mathsf D}_N^{\prime}/{\mathsf D}_N\). Equivalently, \({\mathsf T}_N\) is the logarithmic derivative of \({\mathsf S}_N^{(1)}\).
A different reciprocal family comes from normalizing each exponential section at the origin. The function
is a unit series in \(z^N\). Its powers, after multiplication by \(e^{xz}\), give Appell polynomials. The theta family couples a scaled argument with a reciprocal order.
The zero theory begins only after these algebraic relations have been established. The factorization
turns every orbit of \(N\) zeros in the \(z\) plane into one zero in the \(w\) plane. The large zeros in the \(z\) plane are located by two dominant exponentials. The zeros in the \(w\) plane then give a genus zero product. The logarithm of that product returns to the coefficient theory through
This equality is the bridge between the first and second halves of the paper.
All powers with a complex parameter are first defined as formal series at the origin. To represent this germ analytically, use a simply connected zero free region containing the origin and the logarithm normalized to vanish there. On other regions, specify a branch separately. Integer reciprocal powers are globally meromorphic and do not require a branch choice.
The main statements fall into five groups. The finite algebra model in Section 6 explains the derivative cycles and addition laws through one exponential identity. The coefficient theory in Section 7 gives two complementary recurrences, one from the original denominator and one from its logarithm. The Appell theory in Section 9 supplies both formal differential-operator formulas and a convolution law in the space and order parameters. The zero analysis in Section 11 specializes classical two term exponential-sum theory, uses symmetry to place the uniquely localized large zeros on the rays, and computes a leading correction. The product theory in Section 12 gives the standard diagonal Fredholm encoding of the genus zero product.
The volume keeps formal and analytic statements separate. A formal identity needs only the unit constant term of the relevant series. A holomorphic complex power also needs a zero free domain and a chosen logarithm. A meromorphic coefficient asymptotic is stated only for positive integral reciprocal order unless an additional branch domain is specified.