New identities for the partial Bell polynomials

Understand

A new explicit closed-form formula for the multivariate $(n, k)$th partial Bell polynomial $B_{n,k} (x_1, x_2, ..., x_{n - k + 1})$ 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 $k$ ($n\geq k, k =2, 3,...$).
• Also, a new addition formula (with respect to $k$) 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.