which enumerates the partitions of the multipartite number nn′n″ ... into the parts
abc ..., a′b′c′ ..., a″b″c″ ... ....
Sylvester has determined an analytical expression for the coefficient of xn in the expansion of
| 1 | . |
| (1 - xa) (1 - xb) ... (1 - xi) |
To explain this we have two lemmas:—
Lemma 1.—The coefficient of x-1, i.e., after Cauchy, the residue in the ascending expansion of (1 - ex)-i, is -1. For when i is unity, it is obviously the case, and
| (1 - ex)-i-1 = (1 - ex)-i +ex(1 - ex)-i-1 =(1 - ex)-i + | d | (1 - ex)-i · | 1 | . |
| dx | i |
Here the residue of d/dx (1 - ex)-i · 1/i is zero, and therefore the residue of (1 - ex)-i is unchanged when i is increased by unity, and is therefore always -1 for all values of i.
Lemma 2.—The constant term in any proper algebraical fraction developed in ascending powers of its variable is the same as the residue, with changed sign, of the sum of the fractions obtained by substituting in the given fraction, in lieu of the variable, its exponential multiplied in succession by each of its values (zero excepted, if there be such), which makes the given fraction infinite. For write the proper algebraical fraction
| F(x) = ΣΣ | cλ, μ | + Σ | γλ | . |
| (aμ - x)λ | xλ |