Solutions for Hecke Sum Questions of Banerjee and Bringmann

The present authors introduced a two-color partition series $S(q)$ and conjectured a Hecke-type formula for the even part of $(q^4;q^4)_\infty S(q)$. Banerjee and Bringmann proved the conjecture by using indefinite theta functions, modular completions, and Sturm's theorem. They also asked whether a direct proof, for instance one based on Bailey-type ideas, could be found, and they suggested that the odd residue classes may be worth studying. We prove a two-variable refinement with an additional parameter $a$. Our proof relies entirely on $q$-series combined with the Bailey pairs The original even identity and the odd identity then follow as corollaries by letting $a=1$. We also record parameter symmetries and cyclotomic companions, including a vanishing result at $a=i$.