Recently, Merca defined two q-series: \(\begin{aligned} A_{k,n}(q):&=\sum _{1\le s_1<s_2<\cdots<s_k \le n}\frac{q^{s_1+\cdots +s_k}}{ (1-q^{s_1})^2\cdots (1-q^{s_k})^2},\\ C_{k,n}(q):&= \sum _{1\le s_1< \cdots < s_k \le n}\frac{q^{2(s_1+s_2+\cdots +s_k)-k}}{ (1-q^{2s_1-1})^2(1-q^{2s_2-1})^2\cdots (1-q^{2s_k-1})^2}, \end{aligned}\) which are the truncated forms of MacMahon’s q-series. He also proved some identities involving \(A_{k,n}(q)\) and \(C_{k,n}(q)\) . Very recently, Xia expressed several infinite products in terms of \(A_{k,n}(q) and C_{k,n}(q)\) . In this paper, we prove a number of identities involving \(A_{k,n}(q)\) , \(C_{k,n}(q)\) and partial theta functions via Bailey pair transformations combined with finite expansions due to Xia.
