A direct combinatorial proof of the Jacobi identity

Abstract

An alternate form of the Jacobi identity is equivalent to the assertion that the number of partitions of a Gaussian integer r + si into an odd number of distinct non-zero Gaussian integers p + qi such that i p − qi ≤ 1, p≥0, q≥0, is equal to the number of partitions into an even number of such integers, except when r and s are consecutive triangular numbers. A proof of this assertion is given, based on a dot diagram analogous to that used in Franklin's proof of Euler's theorem relating to the number of partitions of a natural integer into an odd and an even number of distinct parts.