It is well-known that the $q$-Pochhammer symbol has the series expansion $$ \frac{1}{(z;q)} = \sum_{\mathfrak{m} = 0}^{+\infty} \frac{z^m}{(q;q)_\mathfrak{m}} \ . $$
Recently I encountered a double expansion resembling the above, $$ \begin{align} & \ \frac{z^{-1}}{(q_1q_2)^{1/2} - (q_1q_2)^{ - 1/2}} + \sum_{\mathfrak{m}_1, \mathfrak{m}_2 = 0}^{+\infty} \frac{z^{\mathfrak{m}_1 + \mathfrak{m}_2}}{(q_1;q_1)_{\mathfrak{m}_1}(q_2;q_2)_{\mathfrak{m}_2}} R(\mathfrak{m}_1, \mathfrak{m}_2) \\ = & \ \frac{1}{(q_1q_2)^{1/2} - (q_1q_2)^{ -1/2}} \frac{1 - z}{z} \frac{1}{(z;q_1)(z;q_2)} \ , \end{align} $$ where $$ R(\mathfrak{m}_1, \mathfrak{m}_2) = \frac{1}{q_1^{\frac{\mathfrak{m}_1}{2}}q_2^{ - \frac{\mathfrak{m}_2 + 1}{2}} - q_1^{ - \frac{\mathfrak{m}_1}{2}}q_2^{ \frac{\mathfrak{m}_2 + 1}{2}}}\cdot \frac{1}{q_1^{\frac{\mathfrak{m}_1 + 1}{2}}q_2^{ - \frac{\mathfrak{m}_2}{2}} - q_1^{ - \frac{\mathfrak{m}_1 + 1}{2}}q_2^{ \frac{\mathfrak{m}_2}{2}}} \ . $$
I wonder how to prove this? Is it discussed in the literature?