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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-zerocoefficients in the spectral representation where $(N_k)_{k \ge 1}$ is asequence of i.i.d $\mathscr{N}(0,1)$. Using the recently discovered (see Arraset 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 casewhere $F_\infty$ is distributed as the normal product distribution $N_1 \timesN_2$, and relate the associated polynomials class to Rota's {\it Umbralcalculus} by showing that it is a \textit{Sheffer family} and enjoys manyinteresting properties. Lastly, we study the connection between the polynomialclass $\PP_\infty$ and the non-central probabilistic limit theorems within thesecond Wiener chaos.

On a new Sheffer class of polynomials related to normal product distribution