New identities for the partial Bell polynomials

Abstract

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 Bn,k directly for given values of n and k (n GT = k, k = 2, 3,...). Also, a new addition formula (with respect to k) is found for the polynomials Bn,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. (C) 2011 Elsevier Ltd. All rights reserved.