Several $q$-series identities from the Euler expansions of $(a;q)_{\infty }$ and $\frac{1}{(a;q)_{\infty }}$

summary:In this paper, we first give several operator identities which extend the results of Chen and Liu, then make use of them to two $q$-series identities obtained by the Euler expansions of $(a;q)_{\infty }$ and $\frac{1}{(a;q)_{\infty }}$. Several $q$-series identities are obtained involving a $q$-series identity in Ramanujan’s Lost Notebook