By modifying the definition of moments of ranks and cranks, we study the odd moments of ranks and cranks. In particular, we prove the inequality between the first crank moment M 1 (n) and the first rank moment N 1 (n): M 1 (n) > N 1 (n). We also study new counting function ospt(n) which is equal to M 1 (n) − N 1 (n).
Three proofs are given for a reciprocity theorem for a certain q-series found 7 in Ramanujan's lost notebook. The first proof uses Ramanujan's 1 ψ 1 summation the-8 orem, the second employs an identity of N. J. Fine, and the third is combinatorial. 9 Next, we show that the reciprocity theorem leads to a two variable generalization of 10 the quintuple… (More)
Using certain representations for Eisenstein series, we uniformly derive several Ramanujan-type series for 1/π.
In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: Abstract Two new mock theta functions of the sixth order are defined. The main theorem… (More)
A survey of Ramanujan's work on cranks in his lost notebook is given. We give evidence that Ramanujan was concentrating on cranks when he died, that is to say, the final problem on which Ramanujan worked was cranks – not mock theta functions.
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.
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 congru-ences. 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)