A proof of the mod 4 unimodal sequence conjectures and related mock theta functions (Q2078874)

The authors provide detailed background and required basic necessary results needed for this article. A mod \(4\) unimodal sequence conjecture established by \textit{J.Bryson} [``Unimodal sequences and quantum and mock modular forms'', Proc. Natl. Acad. Sci. USA 109, No. 40, 16063--16067 (2012; \url{doi:10.1073/pnas.1211964109})] is proved by the authors. Another mod \(4\) unimodal sequence conjecture given by \textit{B. Kim} et al. [Proc. Am. Math. Soc. 144, No. 9, 3687--3700 (2016; Zbl 1404.11052)] is pointed out as \textit{false}, and suggested its correct form and proof as Theorem 4.1. Authors also established an analogous mod \(4\) results which state as spt-function and for the coefficients of related mock theta function, introduced by \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)]. Several related new results are also discussed in this paper, which are useful for further advancement of the subject.