Infinite families of congruences for c(n) modulo 4 and 8

Establish, for all integers k, n ≥ 0, the following congruences for the coefficients c(m) defined by C(q) = Σ_{m≥0} c(m) q^m with C(q) = q A(q) S(q), A(q) = (-q^2; q^2)_∞ / (q; q^2)_∞^2, and S(q) = Σ_{r≥0} (q; q^2)_r^2 q^{2r} / (-q^2; q^2)_r: c(2^{2k+3} n + (11·4^k + 1)/3) ≡ 0 (mod 4); c(2^{2k+3} n + (17·4^k + 1)/3) ≡ 0 (mod 8); c(2^{2k+4} n + (38·4^k + 1)/3) ≡ 0 (mod 4).

Background

Building on proven cases for k = 0, the authors conjecture three parameterized families of congruences for c(n) modulo 4 and 8 that generalize their established results. These conjectures describe infinite arithmetic progressions in n whose c(n) coefficients vanish modulo 4 or 8.

The proposed families are intended as a roadmap for future research on the congruence properties of coefficients arising from the two-color partition generating function C(q).