Identities from Partition Involutions

Subbarao and Andrews have observed that the combinatorial technique used by F. Franklin to prove Eulers famous partition identity (l-x)(l-x)(l-x)(l-x*) ••• = 1-x-x +x +x -x -x + ••• can be applied to prove the more general formula l-x-xy(l-xy) -xy(±-xy)(±-xy) xy (1 xy) (1 xy) (1 xy) = 1 -x-xy+xy+xy -xy -xy + • •• which reduces to Eulers when y = 1. This note shows that several finite versions of Euler's identity can also be demonstrated using this elementary technique; for example, 1-x-x+x+x-x -x = ( 1 * ) ( 1 ^ 2 ) ( 1 ^ 3 ) ( 1 ^ ) ( 1 X 5 ) ( 1 ; E 6 ) x 7 a x) a x) a x) a x) +x a x) a x*) -x = a ~ x ) a x 2 ) a ~ x 3 ) x k a x 2 ) a x 3 ) + x l > + 5 a x 3 ) x k + 5 + 6 . By using Sylvester*s modification of Franklin's construction, it is also possible to generalize Jacobi's triple product identity. This research was supported in part by National Science Foundation grant MCS 72-03752 A03 and by the Office of Naval Research contract N00014-76-C0330. Reproduction in whole or in part is permitted for any purpose of the United States Government.