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

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 $\mathcal{N}(0,1)$. Using the recently discovered (see Arras et al. \cite{a-a-p-s-stein}) stein operator ${\text{RR}}_{\mathrm{\infty }}$ associated to $F_{\mathrm{\infty }}$, we introduce a new class of polynomials $${\text{PP}}_{\mathrm{\infty }}:=\{P_n={\text{RR}}_{\mathrm{\infty }}^n\mathbf{\text{1}}:n\ge 1\}\mathrm{.}$$ We analysis in details the case where $F_{\mathrm{\infty }}$ 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 $\mathit{\text{Sheffer}}\mathit{\text{family}}$ and enjoys many interesting properties. Lastly, we study the connection between the polynomial class ${\text{PP}}_{\mathrm{\infty }}$ and the non-central probabilistic limit theorems within the second Wiener chaos.