A refined view of a curious identity for partitions into odd parts with designated summands

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are constructed by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In the same work, they also considered the restricted partitions with designated summands wherein all parts must be odd, and they denoted the corresponding function by $\mathrm{PDO}\left(n\right)$.

One of the most curious identities satisfied by $\mathrm{PDO}\left(n\right)$ is the following: For all $n\ge 0$,$\mathrm{PDO}\left(2n\right)=\sum _{k=0}^n\mathrm{PDO}\left(k\right)\mathrm{PDO}(n-k).$ From a generating function perspective, this is equivalent to proving that$\sum _{n\ge 0}\mathrm{PDO}\left(2n\right)q^n={\left(\sum _{n\ge 0}\mathrm{PDO}\right(n\left)q^n\right)}^2.$ The above results are easily proven via elementary generating function manipulations.

In this work, our goal is to extend the above results by providing a two–parameter generalization of the above generating function identity. We do so by considering various natural partition statistics and then utilizing ideas that appear in a 2013 paper of Andrews and Rose which date back a century to groundbreaking work of P. A. MacMahon on natural generalizations of divisors sums.