Generalized Tangent Numbers: Combinatorial Interpretations, Spectral Zeta Values, and Kummer Congruences

The ℤ/nℤ-Fourier decomposition of ex yields n entire functions Enk whose ratios Tnk = Enk/En0 define three families of integer or rational sequences: generalized tangent numbers Bm(k)(n), generalized Euler numbers Am(n), and generalized Bernoulli numbers Dm(n). These are identified with the elementary, complete homogeneous, and power-sum symmetric functions of the spectral parameters via Newton's identities. We prove: The n-snakes theorem: |Bm(k)(n)| counts permutations of [nm+k] with descent set exactly {n, 2n, …, mn}. Spectral zeta values: ζEn0(nl) ∈ ℚ for all l, with a closed formula for the total spectral zeta Zn(n) = n / [(n−1)·(n−1)!] · (1 − 2/C(2n,n)) and cross-component vanishing Zn(m) = 0 for n ∤ m. A generalized Kummer congruence: Bm+T(k)(n) ≡ Bm(k)(n) (mod p) with period T = (pf − 1)/n. The theory of n-irregular primes with computed examples and density estimates. The product identities and exact asymptotics used here are established in a companion paper. At n = 2, every result reduces to the classical Euler–Bernoulli–tangent number framework.