Modulo 8 congruence for c(32n+23)

Prove that for all integers n ≥ 0, the coefficient c(32n+23) is divisible by 8, where c(n) are defined by the generating function 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.

Background

The sequence c(n) arises as the coefficients of the q-series C(q) that is the limit generating function for restricted two-color partitions studied by Andrews and El Bachraoui. In this paper, the authors prove several Ramanujan-type congruences for c(n), including c(8n+4) ≡ 0 (mod 4), c(8n+6) ≡ 0 (mod 8), and c(16n+13) ≡ 0 (mod 4), by relating S(q) to mock theta functions.

Motivated by these results, the authors propose a new congruence modulo 8 for the progression 32n+23, supported by numerical evidence up to 32n+23 ≤ 5000, and present it as an open conjecture for future work.