| 1 | = 1 + | a | + | a² | + ... + | aj | + .... |
| 1 - a. 1 - ax. 1 - ax².... | 1 - x | 1 - x. 1 - x² | 1 - x. 1 - x².... 1 - xj |
The coefficient of ajxn in the expansion of
| 1 |
| 1 - a. 1 - ax. 1 - ax². ... 1 - axi |
denotes the number of ways of composing n with j or fewer parts, none of which are greater than i. The expansion is known to be
| Σ | 1 - xj+1. 1 - xj+2. ... 1 - xj+i | aj. |
| 1 - x. 1 - x². ... 1 - xi |
It has been established by the constructive method by F. Franklin (Amer. Jour. of Math. v. 254), and shows that the generating function for the partitions in question is
| 1 - xj+1. 1 - xj+2. ... 1 - xj+i | , |
| 1 - x. 1 - x². ... 1 - xi |
which, observe, is unaltered by interchange of i and j.
Franklin has also similarly established the identity of Euler
| j = - ∞ | |
| (1 - x)(1 - x²)(1 - x³) ... ad inf. | = Σ (-) jx½(3j²+j), |
| j = + ∞ |