Some extensions of Faà di Bruno’s formula with divided differences

The well-known formula of Faà di Bruno’s for higher derivatives of a composite function has played an important role in combinatorics. In this paper we generalize the divided difference form of Faà di Bruno’s formula and give an explicit formula for the nn-th divided difference of a multicomposite function. More generally, we establish the relationships of the Bell polynomials with respect to multicomposite functions. Applying these to multicomposite functions, we obtain some extensions of Faà di Bruno’s formula.