Analytic proofs of Andrews-Bachraoui identities related to two-color partitions with evens in one color

Andrews and Bachraoui (\textit{Int.

J.

Number Theory} (2026)) studied the two-color partition function $F(n)$ of a non-negative integer $n$ wherein odd parts may appear in two colors (red and blue) and even parts appear in one color (blue).

For any non-negative integer $n,$ they also considered some restricted versions of $F(n)$: $F_0(n)$: the number of partitions of $n$ counted by $F(n)$ such that the number of odd parts in red color is even; $F_1(n)$: the number of partitions counted by $F(n)$ such that the number of odd parts in red color is odd; $H(n)$: the number of partitions of $n$ counted by $F(n)$ such that the parts of the same color do not repeat.

The main purpose of this paper is to present the analytic proofs of the $q$-series identities connected with $F(n)$ and $H(n),$ which appeared as open problems in the original paper.

We also prove some congruences of $F_0(n)$ and $F_1(n)$ modulo $2,$ $4,$ and $8$ by using $ q $-series.