Restricted partition pairs

Abstract

The generating function for sets of pairs of partitions that have an ordering relation between parts of the two pairs is obtained by an inclusion-exclusion argument. Some of the sets of pairs can, by virtue of the extra ordering relation, be interpreted as single partitions. When this is the case an identity between the two generating functions is established. A number of identities including Euler's theorem and the Rogers-Ramanujan identities are obtained by these means in a doubly bounded form. Each doubly bounded identity combines two known but previously unrelated identities into a single one. The generating functions can be summed to produce another of the same type but with different restrictions. By using this summation an infinite number of identities between a single and a multiple summation can be established. © 1993 Academic Press, Inc.