New identities for the partial Bell polynomials

A new explicit closed-form formula for the multivariate (n,k)(n,k)th partial Bell polynomial Bn,k(x1,x2,…,xn−k+1)Bn,k(x1,x2,…,xn−k+1) is deduced. The formula involves multiple summations and makes it possible, for the first time, to easily evaluate Bn,kBn,k directly for given values of nn and kk (n≥k,k=2,3,…n≥k,k=2,3,…). Also, a new addition formula (with respect to kk) is found for the polynomials Bn,kBn,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.