Truncated theta series and a problem of Guo and Zeng
\textit{V. J. W. Guo} and \textit{J. Zeng} [J. Comb. Theory, Ser. A 120, No. 3, 700--707 (2013; Zbl 1259.05020)] proved truncated versions of two classical theta identities of Gauss. These results have inequality-consequences concerning \(\overline{p}(n)\), the number of overpartitions of \(n\), resp. \(\mathrm{pod}(n)\), the number of partitions of \(n\) in which odd parts are not repeated. In the paper under review, the authors give partition-theoretic interpretations of these inequalities by the number of overpartitions of \(n\) in which the first part larger than \(k\) appears at least \(k+1\) times, resp. by the number of partitions of \(n\) in which the first part larger than \(2k-1\) is odd and appears exactly \(k\) times, all other odd parts appear at most once. The authors also prove that the truncated versions in question, together with their previous truncation of Euler's pentagonal number theorem, are essentially corollaries of the Rogers-Fine identity.
scientific article; zbMATH DE number 6804252