Rank, two-color partitions and Mock theta function
George E. Andrews, Rahul Kumar
TL;DR
This work connects partition ranks, two-color partitions, and mock theta functions by proving that the number of partitions of $n$ with positive odd rank equals the number of two-color partitions with a constrained smallest part, and by obtaining a new representation for the mock theta function $f_3(q)$ along with an analogue of the smallest-part function identity. It develops and applies a suite of $q$-series tools, including ${}_{r+1}\phi_r$ transformations and Ramanujan-type summations, to derive generating-function identities linking $f_3(q)$ and $\,\phi_3(q)$ and to establish the equality $N_o^+(n)=G(n)$, where $G(n)$ counts color-partitions with a specified even smallest part. The paper further analyzes the odd-smallest-part variant $G'(n)$, expressing its generating function in terms of $\, phi_3(-q)$ and related theta-series, using analytic continuation to obtain a closed form. Overall, the results illuminate deep connections between partition ranks, color-partitions with part constraints, and third-order mock theta functions, and open avenues for bijective proofs and further study of smallest-part analogues.
Abstract
In this paper, we establish that the number of partitions of a natural number with positive odd rank is equal to the number of two-color partitions (red and blue), where the smallest part is even (say $2n$) and all red parts are even and lie within the interval $(2n,4n]$. This led us to derive a new representation for the third order mock theta function $f_3(q)$ and an analogue of the fundamental identity for the smallest part partition function Spt$(n)$, both of which are of significant interest in their own right. We also consider the odd smallest part version of the above two-color partition, whose generating function involves another third order mock theta function $φ_3(q)$.