Some identities for the partition function
Consider the famous Ramanujan congruences \[ p(5n+4)\equiv 0 \mod 5, \quad p(7n+5)\equiv 0 \mod 7, \quad \text{and} \quad p(11n+6)\equiv 0 \mod 11 \] for the number \(p(n)\) of the partitions of \(n\). Ramanujan obtained \textit{witness} identities between \(q\)-series from which the first two congruences follow as consequences. For the third congruence, \textit{J. Lehner} [Am. J. Math. 65, 492--520 (1943; Zbl 0060.10007)], \textit{A. O. L. Atkin} [Glasg. Math. J. 8, 14--32 (1967; Zbl 0163.04302)], \textit{P. Paule} and \textit{C.-S. Radu} [Springer Proc. Math. Stat. 221, 625--639 (2017; Zbl 1416.11155)] obtained different witness identities. In the paper under review the authors obtain a new witness identity for the third congruence and a new proof of an identity of \textit{H. Zuckerman} [Duke Math. J. 5, 88--110 (1939; JFM 65.0160.02)] for the generating function of \(p(13n+6)\). They also establish some new identities for the generating functions for \(p(17n+5)\), \(p(19n+4)\), and \(p(23n+1)\).
scientific article; zbMATH DE number 7451091
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