6 problems

Bell-polynomial identity. For any $l,n_0,\dots,n_p\in\mathbb{N}$, the Bell polynomials satisfy the identity

Generalized Bell-polynomial relation. The relation

Let $\gamma_j$ be real numbers for $0\leq j\leq K$, with $\gamma_0=1$, and define … Define … Let $p$ be the monic polynomial of degree $K$ satisfying…

Let $B(k,\mu)=\sum_i {k\brace i}\mu^i$ denote the Bell polynomial, and write $B_x=B(x,1)$. For a Poisson random variable with mean $\mu$, its $k$th raw moment is $B(k,\mu)$. Bell-n…

Let $p$ be a fixed positive integer. Let $u_1=1$, let $u_2,\dots,u_p$ and $d_1,\dots,d_p$ be variables, and let $F_1=1$ and $F_i=F_i(\{u_k\},\{d_k\})$ for $i=2,\dots,p$ be the uniq…

Let $d_n=\sum_{j=1}^\infty a_jw_j^n$, where the magnitudes $|w_j|$ are strictly decreasing and $|a_j|>1$. Let $B_n(d)$ denote the complete exponential Bell polynomial evaluated at…