题目: Strongly unimodal sequences and Hecke-type identities
报告人: 崔素平 (青海师范大学)
时间：2023年11月13日 （星期一）9:30-10:30
地点： 腾讯会：754-139-165
腾讯会议链接 //meeting.tencent.com/dm/mxi3N7M6ih3J
摘要：A strongly unimodal sequence of size $n$ is a sequence of integers $\{a_j\}_{j=1}^s$ 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 $m$, while the generating function of $u(m,n)$ is written as $$\mathcal{U}(z;q) := \sum_{m,n} u(m,n)z^mq^n.$$
Recently, Chen and Garvan established some Hecke-type identities for the third order mock theta function $\psi(q)$ and $U(q)$, which are the specializations of $\mathcal{U}(z;q)$, as advocated by $\psi(q)=\mathcal{U}(\pm i;q)$ and $U(q)=\mathcal{U}(1;q)$. 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. Our work is built upon a handful of known and newly constructed Bailey pairs and conjugate Bailey pairs.
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邀请人：毛仁荣