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 e^x onto residue classes modulo n via the discrete Fourier transform (DFT) yields a canonical basis E_n^k(x). For n = 2, this recovers the classical hyperbolic functions cosh and sinh.

This paper develops the algebraic theory of these functions systematically, focusing on three structural pillars:

1. Convolution Algebra: The vector of functions (E_n^0, ..., E_n^(n-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.

2. Circulant Matrices & Pythagorean Identities: The associated n x n circulant matrix C_n(x) has a determinant of 1 for all x. Expanding this determinant produces multilinear generalizations of the Pythagorean identity cosh^2 - sinh^2 = 1.

3. Generalized Tangent Functions: We define the tangent functions T_n^k = E_n^k / E_n^0 and derive their closed autonomous ODE system.

Additionally, the paper establishes an additive-multiplicative trace identity connecting the cyclic and scaling structures on Z/nZ, providing a determinantal formula for the Legendre symbol.