1 Introduction
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For a power series
and a positive integer \(N\), the coefficients may be separated according to the residue of \(m\) modulo \(N\). Root of unity averaging gives exact projection operators for this separation. This elementary construction is usually called a multisection or a root filter. It appears throughout the classical theory of generating functions and special functions. Background on multisections, hypergeometric functions, and related coefficient systems may be found in [Rainville1960], [AndrewsAskeyRoy1999], [Comtet1974].
Papers I and II developed typed seed grammars, exact projector packets, and finite residue-projector algebra, while Paper III treated typed edge operators and exact coordinate transport [PaperI2026], [PaperII2026], [PaperIII2026]. The root filter used below is the finite cyclic scalar specialization of that earlier projector calculus. The signed companion matrix is likewise a concrete cyclic instance, not a replacement for Paper III's general typed-edge theory.
This volume follows the consequences of applying the cyclic projections to the exponential function. The projected exponentials form a finite differential system. After one sign is inserted around the derivative cycle, they form a basis for
The section with initial value one is
Its reciprocal powers produce polynomial coefficient families. Its logarithmic derivative produces a scalar higher Riccati identity. The same reciprocal construction applied to every exponential section gives normalized residue section functions and ordinary Appell polynomial sequences. These cyclic exponential sections and their matrix addition laws belong to the generalized-hyperbolic-function literature [Schwaiger1992], [Muldoon2005]. Appell and binomial-type sequences are discussed in [Roman1984], [Comtet1974].
The second half of the paper concerns the zeros of \({\mathsf D}_N\). The finite exponential sum
has a regular polygon of exponents. The general relation between an exponent polygon and the zero strips of an exponential sum goes back to Pólya and Langer. Later developments include the work of Moreno and the treatment in Levin's monograph [Polya1920], [Langer1931], [Moreno1973], [Levin1980]. Bleher and Mallison give a particularly close general result: for convex-polygon exponential sums they locate all sufficiently large zeros in two term strips and prove exponential localization and simplicity [BleherMallison2006]. The regular polygon and conjugation symmetry of the present family specialize that theory to \(N\) exact rays. We give a self-contained specialization because its notation is used throughout the product and zeta theory, and we compute the leading displacement term for this family. The resulting counting formulas have explicit leading constants and bounded error.
The rotational symmetry is more useful after the change of variable \(w=z^N\). There is an entire function
with
For \(N\geq2\), the order of \(\Delta_N\) is \(1/N\). Its Hadamard product therefore has genus zero. The logarithm of this product identifies the transfer tangent coefficients with power sums of the reduced zeros. The large zero formula also compares the reduced zero zeta with a Hurwitz zeta function. Standard material on entire functions, canonical products, and exponential sums appears in [Boas1954], [Levin1980], [Titchmarsh1939]. We use singularity analysis in the form described in [FlajoletSedgewick2009] and the classical continuation of the Hurwitz zeta function from [Apostol1976].
The cyclic exponential sections and their matrix addition laws have a classical literature. The tools used below include Appell and binomial type sequences, canonical products, singularity analysis, and Fredholm determinants. These tools are also standard. Our aim is to organize this symmetric specialization and link its reciprocal coefficients to reduced-zero data. No historical priority is asserted for the component constructions.