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Operationally, using the generalized Dobinski formula in the MO-Q on the o.g.f. for the Bell polynomials $\phi_m(x) = \sum_{k=0}^m S(m,k) x^k$,

$$ \phi_{n+m}(x) = (\phi.(x))^{n+m} = e^{-x} \sum_{j =0}^{\infty} j^{n+m} \frac{x^j}{j!} = \sum_{j=0}^{\infty} (-1)^j [\sum_{k=0}^j (-1)^k \binom{j}{k} k^{n+m}] \frac{x^j}{j!} \; . $$

Equivalently,

$$ \phi_{p}(\widehat{xD_x})= \sum_{k=0}^p S(p,k)x^kD_x^k = (xD_x)^p=\sum_{j=0}^\infty j^p \frac{x^jD^j_{x=0}}{j!}=\sum_{j=0}^\infty (-1)^j [\sum_{k=0}^j(-1)^k \binom{j}{k}k^p] \frac{x^jD_x^j}{j!} \; ,$$

where $D_x = \frac{d}{dx}$.

Operationally, using the generalized Dobinski formula in the MO-Q on the o.g.f. for the Bell polynomials $\phi_m(x) = \sum_{k=0}^m S(m,k) x^k$,

$$ \phi_{n+m}(x) = (\phi.(x))^{n+m} = e^{-x} \sum_{j =0}^{\infty} j^{n+m} \frac{x^j}{j!} = \sum_{j=0}^{\infty} (-1)^j [\sum_{k=0}^j (-1)^k \binom{j}{k} k^{n+m}] \frac{x^j}{j!} \; . $$

Equivalently,

$$ \phi_{p}(\widehat{xD_x})= \sum_{k=0}^p S(p,k)x^kD_x^k = (xD_x)^p=\sum_{j=0}^\infty j^p \frac{x^jD^j_{x=0}}{j!}=\sum_{j=0}^\infty (-1)^j [\sum_{k=0}^j(-1)^k \binom{j}{k}k^p] \frac{x^jD_x^j}{j!} \; ,$$

where $D_x = \frac{d}{dx}$.

Operationally, using the generalized Dobinski formula in the MO-Q on the o.g.f. for the Bell polynomials $\phi_m(x) = \sum_{k=0}^m S(m,k) x^k$,

$$ \phi_{n+m}(x) = (\phi.(x))^{n+m} = e^{-x} \sum_{j =0}^{\infty} j^{n+m} \frac{x^j}{j!} = \sum_{j=0}^{\infty} (-1)^j [\sum_{k=0}^j (-1)^k \binom{j}{k} k^{n+m}] \frac{x^j}{j!} \; . $$

Equivalently,

$$ \phi_{p}(\widehat{xD_x})= \sum_{k=0}^p S(p,k)x^kD_x^k})^k = (xD_x)^p=\sum_{j=0}^\infty j^p \frac{x^jD^j_{x=0}}{j!}=\sum{j=0}^\infty (-1)^j [\sum_{k=0}^j](-1)^k binom{j{k}k^p] \frac{x^jD_x^j}{j!}$$

Operationally, using the generalized Dobinski formula in the MO-Q on the o.g.f. for the Bell polynomials $\phi_m(x) = \sum_{k=0}^m S(m,k) x^k$,

$$ \phi_{n+m}(x) = (\phi.(x))^{n+m} = e^{-x} \sum_{j =0}^{\infty} j^{n+m} \frac{x^j}{j!} = \sum_{j=0}^{\infty} (-1)^j [\sum_{k=0}^j (-1)^k \binom{j}{k} k^{n+m}] \frac{x^j}{j!} \; . $$

Equivalently,

$$ \phi_{p}(\widehat{xD_x})= \sum_{k=0}^p S(p,k)x^kD_x^k = (xD_x)^p=\sum_{j=0}^\infty j^p \frac{x^jD^j_{x=0}}{j!}=\sum_{j=0}^\infty (-1)^j [\sum_{k=0}^j(-1)^k \binom{j}{k}k^p] \frac{x^jD_x^j}{j!} \; ,$$

where $D_x = \frac{d}{dx}$.

Operationally, using the generalized Dobinski formula in the MO-Q on the o.g.f. for the Bell polynomials $\phi_m(x) = \sum_{k=0}^m S(m,k) x^k$,

$$ \phi_{n+m}(x) = (\phi.(x))^{n+m} = e^{-x} \sum_{j =0}^{\infty} j^{n+m} \frac{x^j}{j!} = \sum_{j=0}^{\infty} (-1)^j [\sum_{k=0}^j (-1)^k \binom{j}{k} k^{n+m}] \frac{x^j}{j!} \; . $$

Operationally, using the generalized Dobinski formula in the MO-Q on the o.g.f. for the Bell polynomials $\phi_m(x) = \sum_{k=0}^m S(m,k) x^k$,

$$ \phi_{n+m}(x) = (\phi.(x))^{n+m} = e^{-x} \sum_{j =0}^{\infty} j^{n+m} \frac{x^j}{j!} = \sum_{j=0}^{\infty} (-1)^j [\sum_{k=0}^j (-1)^k \binom{j}{k} k^{n+m}] \frac{x^j}{j!} \; . $$

Equivalently,

$$ \phi_{p}(\widehat{xD_x})= \sum_{k=0}^p S(p,k)x^kD_x^k})^k = (xD_x)^p=\sum_{j=0}^\infty j^p \frac{x^jD^j_{x=0}}{j!}=\sum{j=0}^\infty (-1)^j [\sum_{k=0}^j](-1)^k binom{j{k}k^p] \frac{x^jD_x^j}{j!}$$