An algebraic identity of F. H. Jackson and its implications for partitions (Q5937447)

The authors use the identity NEWLINE\[NEWLINE1- {a(1- b)(1- c)(1- d)(1- a^2 bcd)\over (1- ab)(1- ac)(1- ad)(1- abcd)}= {(1- a)(1- abc)(1- abd)(1- acd)\over (1- ab)(1- ac)(1- ad)(1- abcd)}NEWLINE\]NEWLINE established long ago by \textit{F. H. Jackson} [Messenger 50, 101-121 (1920; JFM 48.0415.01)] to derive generating functions for certain restricted partitions. This approach enables them to obtain also several identities related to the newly defined partition functions. In the concluding part of the paper they remark that bijective proofs of their theorems would be of great interest and outline possible extensions and generalizations.