Factorization of Lagrange’s Expansion by Means of Exponential Generating Functions

The direct expansion of the individual terms in (2) and (3) involves repeated use of combinatorial algebra with restrictions on the indices; the resulting coefficients, apart from one factor, are of the form corresponding to the factorization of the exponential of a series of derivatives. This was first found by Sylvester [18] for the inversion of power series; the same feature is shown by the corresponding expressions obtained by Mellin [13], Birkeland [3] and Frame [8] in terms of generalized hypergeometric series, and by the expansions for the simultaneous inversion of several power series [5], [19]. This similarity to the exponential series suggests that it may be possible to separate off, from the start, those factors which do not agree with the exponential, and to use the remaining factors as coefficients of a rigorously