Infinite family of congruence conjectures for c(n)

Prior proof and attribution. All three conjectured families were already proved by Sun and Yao in arXiv:2604.05403, Section 2. The original problem is therefore not open. The proof below is only an alternative derivation through a stronger residue-three transfer identity; it does not claim the first proof.

All three congruence families hold for every $k,n\ge0$. The key is a transfer theorem between the source's new coefficient sequence $c$ and the distinct mock-theta partition function $p_\omega$:

$$\boxed{ c(4m+3)\equiv-p_\omega(4m+3)\pmod8 \qquad(m\ge0), }$$

where $q\omega(q)=\sum_{N\ge1}p_\omega(N)q^N$.

Indeed, the primary source gives

$$C(q)=\sum_{N\ge0}c(N)q^N = 2qA(q)B(-q)-q\omega(-q),$$

and

$$A(q)\equiv A_0(q^4)+qA_1(q^4)\pmod4, \qquad A_1\in2\mathbb Z[[q]].$$

For $B(q)=\sum_{j=0}^3q^jB_j(q^4)$, the exact classical four-dissections are

$$B_1(q)=\frac{2f_2^8}{f_1^7}, \qquad B_2(q)=\frac{4f_2^2f_4^4}{f_1^5}, \qquad f_j=(q^j;q^j)_\infty.$$

Thus $B_1$ is even and $B_2$ is divisible by $4$. Consequently the residue-two part of $A(q)B(-q)$ satisfies

$$A_0B_2-A_1B_1\equiv0\pmod4.$$

Therefore $2qA(q)B(-q)$ contributes zero modulo $8$ to every exponent $4m+3$, while

$$-[q^{4m+2}]\omega(-q) = -p_\omega(4m+3),$$

proving the transfer theorem.

Now define

$$\begin{aligned} N_1(k,n) &= 2^{2k+3}n+\frac{11\cdot4^k+1}{3},\\ N_2(k,n) &= 2^{2k+3}n+\frac{17\cdot4^k+1}{3},\\ N_3(k,n) &= 2^{2k+4}n+\frac{38\cdot4^k+1}{3}. \end{aligned}$$

For every $k\ge1$, all three integers satisfy $N_i(k,n)\equiv3\pmod4$. The published theorem of Andrews–Passary–Sellers–Yee gives

$$p_\omega(N_1(k,n))\equiv0\pmod4,$$ $$p_\omega(N_2(k,n))\equiv0\pmod8,$$

and

$$p_\omega(N_3(k,n))\equiv0\pmod4.$$

Applying the transfer identity proves the corresponding three congruences for $c$ for every $k\ge1$.

When $k=0$, the three statements reduce to

$$c(8n+4)\equiv0\pmod4, \qquad c(8n+6)\equiv0\pmod8, \qquad c(16n+13)\equiv0\pmod4,$$

which are already established as Theorems 1.4–1.6 in the primary source. Hence all three conjectured families hold for every $k,n\ge0$.

The 2017 theorem concerns $p_\omega$, not $c$; Sun and Yao already proved the stated conjecture in arXiv:2604.05403, and the transfer identity gives an alternative stronger route.

Sources: Banerjee–Bringmann–El Bachraoui, arXiv:2604.02239, Conjecture 8.2; Andrews–Passary–Sellers–Yee, The Ramanujan Journal 43 (2017), 347–357, Theorem 1.1, doi:10.1007/s11139-016-9812-2; “On second order mock theta function $B(q)$,” Electronic Research Archive 30 (2022), 52–65, equations (1.6)–(1.7), doi:10.3934/era.2022003.