  Consider a generic random element $F_\infty= \sum_{\text{finite}} \lambda_k
(N^2_k -1)$ in the second Wiener chaos with a finite number of non-zero
coefficients in the spectral representation where $(N_k)_{k \ge 1}$ is a
sequence of i.i.d $\mathscr{N}(0,1)$. Using the recently discovered (see Arras
et al. \cite{a-a-p-s-stein}) stein operator $\RR_\infty$ associated to
$F_\infty$, we introduce a new class of polynomials $$\PP_\infty:= \{ P_n =
\RR^n_\infty \textbf{1} \, : \, n \ge 1 \}.$$ We analysis in details the case
where $F_\infty$ is distributed as the normal product distribution $N_1 \times
N_2$, and relate the associated polynomials class to Rota's {\it Umbral
calculus} by showing that it is a \textit{Sheffer family} and enjoys many
interesting properties. Lastly, we study the connection between the polynomial
class $\PP_\infty$ and the non-central probabilistic limit theorems within the
second Wiener chaos.
