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

Abstract

We provide a simple proof of a partial theta identity of Andrews and study the underlying combinatorics. This yields a weighted partition theorem involving partitions into distinct parts with smallest part odd which turns out to be a companion to a weighted partition theorem involving the same partitions that we recently deduced from a partial theta identity in Ramanujan's Lost Notebook. We also establish some new partition identities from certain special cases of Andrews' partial theta identity.

Some Combinatorial and Analytical Identities

“…In a recent paper Chen and Liu [7] generalized earlier work of Alladi [1] involving weighted partition theorems. Chen and Liu were interested in Franklin type involutions, and as an application of their technique, they proved the identity…”

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

“…Andrews recently showed me that (6.2) can be refined to In a subsequent paper [1] we will analyze (6.3) combinatorially, show that it yields a weighted partition theorem which is a companion to Theorem 1, and deduce some new weighted partition identities as well.…”

The leading root of the partial theta function

“…12 From (5.1) we then have which differs from the exact ξ 0 (y) −2 starting at order y 8 . The difference at order y 8 arises from a contribution to ξ 0 (y) that is proportional to a 2 (y) 2 . The full structure of the contributions to ξ 0 (y) and its powers can be read off the explicit implicit function formula [44]: see [47] for details.…”

“…2. It would be interesting to seek a combinatorial interpretation of the coefficients of 1 − 1/ξ 0 (y) 2 , analogously to what Prellberg [37] has done for ξ 0 (y) [see footnotes 9-11 above].…”

“…a n−1 a n+1 ≥ a 2 n for all n ≥ 1. A classic theorem of Kaluza [27] relates Conjecture 1.6 to Theorem 1.2: namely, if the coefficient sequence (a n ) ∞ n=0 of a formal power series f is strictly positive and log convex, then 1/f has nonpositive coefficients after the constant term; and if in addition a 0 a 2 > a 2 1 , then 1/f has strictly negative coefficients after the constant term. 2 But it is easily seen that the converse does not hold.…”