Strongly unimodal sequences and Hecke-type identities

Abstract

A strongly unimodal sequence of size n is a sequence of integers ${\left\{a_j\right\}}_{j=1}^s$ satisfying the following conditions:$0<a_1<a_2<\cdots <a_k>a_{k+1}>\cdots >a_s>0\text{and}{a}_1+a_2+\cdots +a_s=n,$ for a certain index k, and we usually define its rank as $s-2k+1$. Let $u(m,n)$ be the number of strongly unimodal sequences of size n with rank m, and the generating function for $u(m,n)$ is written as$U(z;q):=\sum _{m,n}u(m,n)z^m{q}^n.$ Recently, Chen and Garvan established some Hecke-type identities for the third order mock theta function $\psi \left(q\right)$ and $U\left(q\right)$, which are the specializations of $U(z;q)$, as advocated by $\psi \left(q\right)=U(\pm i;q)$ and $U\left(q\right)=U(1;q)$. Meanwhile, they inquired whether these Hecke-type identities could be proved via the Bailey pair machinery. In this paper, we not only answer the inquiry of Chen and Garvan in the affirmative, but offer more instances in a broader setting, with, for example, some classical third order mock theta functions due to Ramanujan involved. Furthermore, we extend the Hecke-type identities into multiple series identities. Our work is built upon a handful of Bailey pairs and conjugate Bailey pairs.