Appendix B: Applications of the Lemma

 If we apply lemma (30) to Chebyshev-Hermite polynomials in (7), we get g(x)=-x²/2 for $f=\exp$, so that only the terms with m=1 and m=2 are non-zero in the product in (30), and we only need non-negative integers $\{k_1, k_2\}$ as the solutions for k₁+2k₂=n. Thus for each $k\equiv k_2$ running from to [n/2] (entier of n/2) we have k₁=n-2k and r=n-k. Finally, we have from (30) that

$$\begin{eqnarray} \lefteqn{ {{\rm d}^n \over {\rm d}x^n}\exp(-x^2/2) = n! \sum_... ...n-2k} \ {1 \over k!} \left({1 \over 2!} (-1)\right)^{k} \; , } \end{eqnarray}$$ (B1)

and the explicit expression (13) follows immediately from Rodrigues' formula (7).

Among other consequences of (30) is the relation (32) between cumulants $\kappa_n$ and moments $\alpha_k$ of a PDF. From the definition (29) we obtain this relation by simply applying (30) to the case of $f\equiv \ln$ and $g\equiv\Phi$. Since $f^{(r)}(y)\vert _{y=g(t)}=(-1)^{r-1}(r-1)!/\Phi^r\vert _{t=0}=(-1)^{r-1}(r-1)!$, we find that

$$\begin{eqnarray} \lefteqn{ \kappa_n = {1\over i^n} {{\rm d}^n \over {\rm d}t^n}... ...!} \left({1 \over m!} \Phi^{(m)}\vert _{t=0}\right)^{k_m} \; .} \end{eqnarray}$$ (B2)

Thus, from (26),  

$$\kappa_n = {n!\over i^n} \sum_{\{k_m\}}(-1)^{r-1}(r-1)! \p... ..._m} \over k_m!} \left({\alpha_m \over m!} \right)^{k_m} \; ,$$ (B3)

which is equivalent to (32). Here the sum extends over all non-negative integers $\{k_m\}$satisfying (31) and r=k₁+k₂+ ... +k_n .