ZEROS OF GENERALIZED EXPONENTIAL FUNCTIONS: UNIVERSAL RAY STRUCTURE, ORBIT-COLLAPSED HADAMARD PRODUCTS, AND SPECTRAL SUM RULES

For n ≥ 2, the ℤ/nℤ-Fourier decomposition of ex yields n entire functions Enk(x) = ∑m≥0 xnm+k/(nm+k)! (0 ≤ k ≤ n−1), forming a basis for the ODE y(n) = y with derivative shift (Enk)′ = Enk−1 and Wronskian 1. We study the zeros of all n components simultaneously and the spectral zeta functions ζEnk(s) = ∑z≠0, Enk(z)=0 |z|−s. For odd n ≥ 3, the nonzero zeros of the n components interleave on the negative real axis in perfect cyclic order k = 0, 1, …, n−1, governed by Rolle's theorem and the derivative shift. Each reduced function Fnk = Enk/(xk/k!) admits an orbit-collapsed Hadamard product Fnk(x) = ∏j(1 + xn/rjn), valid uniformly for all n ≥ 2 and 0 ≤ k ≤ n−1. From this factorization we prove the multi-component spectral identity: for all n ≥ 2 and 0 ≤ k ≤ n−1, ζEnk(n) = 1 / [(n−1)! · C(n+k, k)], exhibiting a binomial decay ζEnk(n) / ζEn0(n) = 1/C(n+k, k) across the decomposition. We determine the asymptotic zero locations rj ∼ (2j+1)π / (2 sin(π/n)) with exponentially small error, establish the zero counting function Nk(R) = (sin(π/n)/π) R + O(1) (identical for all components), and prove that the n components divide each single-component gap into n equal parts of width π/(n sin(π/n)). At n = 2 the entire theory reduces to the classical framework of cosh, sinh, and their zeros.