Sign Patterns in a Two Colored Partition Companion series

Abstract

We study three closely related questions arising from the recent work of Andrews and El Bachraoui on the two-color partition series \[ S_1(q)=\sum_{n\ge0}s_1(n)q^n=\sum_{a\ge0}q^a(-q^{a+1};q)_\infty^2 \] and its odd companion, denoted by $T_o(q)$. First, for the eta-normalized companion \[ C(q)=(q;q)_\infty T_o(q)=\sum_{n\ge0}c(n)q^n, \] we prove a strong form of the Andrews--El Bachraoui sign conjecture that $\mathrm{limsup}c(n)=+\infty$ and $\mathrm{liminf}c(n)=-\infty$. Second, we construct an involution using the Franklin-type involution of Chen and Liu to combinatorially explain Andrews--El Bachraoui congruence for $s_1(n)$ modulo 4. Third, we introduce a two-variable refinement of $S_1(q)$ which records the difference of the lengths of the two colored components. Using Ramanujan's partial theta identity and heat-type equations, we obtain a closed formula for all of its even moments, and we show that the eta-normalized moments satisfy termwise congruences modulo $p^{2k}$ for every prime $p\equiv 5(\mathrm{mod}6)$.

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