We establish a set of two new relationships involving $R_{\alpha},R_{\beta}$ and $R_m$-functions, which are based up Jacobi's famous triple-product identity. We, also provide answer for an open problem of Srivastava, Srivastava, Chaudhary and Uddin, which suggest to find an inter-relationships between $R_{\alpha},R_{\beta}$ and $R_m(m\in\mathbb{N})$, $q$-product identities and continued-fraction identities.