GENERALIZED EXPONENTIAL FUNCTIONS : CONVOLUTION ALGEBRA, CIRCULANT MATRICES, AND GENERALIZED PYTHAGOREAN IDENTITIES

The ordinary differential equation y(n) = y admits n linearly independent solutions. Projecting the exponential function ex onto residue classes modulo n via the discrete Fourier transform yields a canonical basis Enk(x) = (1/n) ∑j=0n−1 ω−jk eωjx, ω = e2πi/n, indexed by k ∈ ℤ/nℤ; for n = 2 this recovers cosh and sinh. We develop the algebraic theory of these functions systematically. The vector (En0, …, Enn−1) is shown to be a group homomorphism from (ℝ, +) into the group of units of the group ring ℂ[ℤ/nℤ] equipped with cyclic convolution. The associated n × n circulant matrix Cn(x) lies in SLn(ℂ) for all x; expanding det Cn(x) = 1 produces multilinear generalizations of the classical identity cosh2x − sinh2x = 1. We establish divisibility and tensor decomposition results via the Chinese Remainder Theorem, introduce the generalized tangent functions Tnk = Enk/En0 and their closed autonomous ODE system (Tnk)′ = Tnk−1 − Tnk Tnn−1, prove the closed form B1(k)(n) = 1 − C(n+k, k) for the first generalized tangent numbers and tabulate Bm(k)(n) for n ≤ 11, prove an additive-multiplicative trace identity tr(Cn(x) · Pa) = tr(Cd(x)) where d = gcd(1−a, n), and derive a determinantal formula for the Legendre symbol via Zolotarev's lemma.