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Journals and Conferences
have motivated much research. Here, p(n) denotes the number of partitions of n. In particular, toward a combinatorial explanation of the above congruences many partition statistics have been studied. Among them, the rank suggested by F. Dyson  and the crank suggested by the first author and F.G. Garvan  have proven successful and their own properties… (More)
In this article, we construct a general series for 1 π . We indicate that Ramanujan’s 1 π −series are all special cases of this general series and we end the paper with a new class of 1 π −series. Our work is motivated by series recently discovered by Takeshi Sato.
In his lost notebook, Ramanujan offers several results related to the crank, the existence of which was first conjectured by F. J. Dyson and later established by G.E.Andrews and F.G. Garvan. Using an obscure identity found on p. 59 of the lost notebook, we provide uniform proofs of several congruences in the ring of formal power series for the generating… (More)
In 2007 George E. Andrews and Peter Paule  introduced a new class of combinatorial objects called broken k-diamonds. Their generating functions connect to modular forms and give rise to a variety of partition congruences. In 2008 Song Heng Chan proved the first infinite family of congruences when k = 2. In this note we present two non-standard infinite… (More)
We show how Rank-Crank type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank-Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized… (More)
PURPOSE To examine whether there is an association between students self-reported suicide attempts and non-suicidal self-injury and exposure to suicidal behaviour among friends, family members or within school communities. METHODS A cross-sectional analysis of a nationally representative health and well-being survey of 8500 New Zealand high school… (More)
Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s 1ψ1 summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity.… (More)
In 2003, Atkin and Garvan initiated the study of rank and crank moments for ordinary partitions. These moments satisfy a strict inequality. We prove that a strict inequality also holds for the first rank and crank moments of overpartitions and consider a new combinatorial interpretation in this setting.