Rogers-Ramanujan Type $q$-Exponential Operator and Stieljes-Widgert Polynomials

In this paper, we use the Rogers-Ramanujan type $q$-exponential operator $\mathcal{R}(q{D}_q)$ to derive generating functions, and Mehler and Rogers formulas, for the non-normalized homogeneous Stieljes-Wigert polynomials ${\mathrm{S}}_n(x,y;q)$.