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

Zhizheng Zhang1,2 and Jizhen Yang1

1 Center of Combinatorics and LPMC, Nankai University Tianjin 300071, P. R. China
2 Department of Mathematics, Luoyang Normal University Luoyang 471022, P. R. China


Abstract: 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.

AMSclassification: primary 05A30; secondary 33D15, 33D60.

Keywords: exponential operator, operator identity, q-series identity.