A partial theta identity of Ramanujan and its number-theoretic interpretation

Abstract

We will interpret a partial theta identity in Ramanujan's Lost Notebook as a weighted partition theorem involving partitions into distinct parts with smallest part odd. A special case of this yields a new result on the parity of the number of parts in such partitions, comparable to Euler's pentagonal numbers theorem. We will provide simple and novel proofs of the weighted partition theorem and the special case. Our proof leads to a companion to Ramanujan's partial theta identity which we will explain combinatorially.

A combinatorial study and comparison of partial theta identities of Andrews and Ramanujan

“…It was shown in [3] that even though the series on the left in (6.1) is different from the one on the left in (1.1), the combinatorial interpretation of both identities are the same, namely Theorem 1. In a similar sense, identities (1.6) and (5.1) are companions because they both are analytic representations of Theorem 2.…”

“…Theorem 3 which can be compared to Euler's celebrated Pentagonal Number's Theorem was first observed (and proved) in [3], although Andrews [5] had previously noticed a result closely resembling Theorem 3 (see [3] for a comparison of Andrews' theorem and our Theorem 3).…”

“…In [3] we show how Theorem 3 is equivalent to Andrews' theorem, in the sense that each can be derived from the other. The idea is similar to how we showed above that in (1.6) the real contribution comes from partitions in P d,o and not from partitions in P d,e .…”

“…In [9] a combinatorial proof of (1.1) was given by interpreting the left hand side in terms of vector partitions. In [3] we gave a very down to earth q-series proof of (1.1) without recourse to Heine's transformation. We shall now give a similar proof of (1.6).…”

“…In a recent paper [3] we showed that the partial theta identity Here and throughout, σ (π) is the sum of the parts of π and P d,o is the set of partitions into distinct parts with smallest part odd. Also in (1.1) we have used the standard notation (1 − aq j ), (1.5) when |q| < 1.…”