Euler's classic partition identity states that the number of partitions of n into odd parts equals the number of partitions of n into distinct parts. We develop a new generalization of this identity, which yields a previous generalization of Franklin as a special case, and provide both a q-series and bijective proof. We further establish an accompanying Beck-type companion identity which gives the excess in the total number of parts of partitions of one kind over the other.

This is joint work with Gabriel Gray, David Hovey, Brandt Kronholm, Emily Payne, and Ren Watson.