New identities for the partial Bell polynomials

A new explicit closed-form formula for the multivariate $(n,k)$th partial Bell polynomial $B_{n,k}(x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_{n-k+1})$ is deduced. The formula involves multiple summations and makes it possible, for the first time, to easily evaluate $B_{n,k}$ directly for given values of $n$ and $k$ ($n\geq k, k =2, 3,…$)\mathrm{.}Also,anewadditionformula(withrespectto$k$) is found for the polynomials $B_{n,k}$ and it is shown that they admit a new recurrence relation. Several special cases and consequences are pointed out, and some examples are also given.