Abstract:
Ramanujan recorded five interesting q-series identities in a section that is not as systematically
arranged as the other chapters of his second notebook. These five identities
do not seem to have acquired enough attention. Recently, Dixit and the third author
found a one-variable generalization of one of the aforementioned five identities. From
their generalized identity, they were able to derive the last three of these q-series identities
but did not establish the first two. In the present article, we derive a one-variable
generalization of the main identity of Dixit and the third author from whichwe successfully
deduce all the five q-series identities of Ramanujan. In addition to this, we also
establish a few interesting weighted partition identities from our generalized identity.
In the mid 1980s, Bressoud and Subbarao found an interesting identity connecting the
generalized divisor function with a weighted partition function, which they proved by
means of a purely combinatorial argument. Quite surprisingly, we found an analytic
proof for a generalization of the identity of Bressoud and Subbarao, starting from the
fourth identity of the aforementioned five q-series identities of Ramanujan.