General Colored Partition Identities

Ramanujan’s modular equations of degrees 3, 5, 7, 11 and 23 yield beautiful colored partition identities. Warnaar analytically generalized the modular equations of degrees 3 and 7, and thereafter, the author found bijective proofs of those partitions identities and recently, established an analytic generalization of the modular equations of degrees 5, 11 and 23. The partition identities of degrees 5 and 11 were combinatorially proved by Sandon and Zanello, and it remains open to find a combinatorial proof of the partition identity of degree 23. In this paper, we prove general colored partition identities with a restriction on the number of parts, which are connected to the partition identities arising from those modular equations. We also provide bijective proofs of these partition identities. In particular, one of these proofs gives bijective proofs of the partition identity of degree 23 for some cases, which also work for the identities of degrees 5 and 11 for the same cases.