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 , … , 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 ≥ 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.