Identities involving partitions with distinct odd parts and no parts congruent to 2 mod 4

Abstract

Recently, Pankaj Jyoti Mahanta and Manjil P. Saikika proved some identities relating certain restricted partitions into distinct odd parts with the partition whose odd parts are distinct combinatorially. They asked for the q-series proofs. In this paper, we give q-series proofs of these identities. Also, the number of partitions into distinct odd parts equals to the number of the partitions with no parts congruent to 2 mod 4, so we can get some identities. We also give combinatorial proofs of these identities.