Development of Partition Functions of Ramanujan’s Works

In 1986, Dyson defined the rank of a partition as the largest part of a partition p minus the number of parts of p . In1988, Garvan discussed the theta series in x like A(x), B(x), C(x), D(x) and also discussed Jacobi’s triple product Identity(1829). Both of the authors have worked on Ramanujan’s seminal works Ramanujan’s Lost Notebooks . This paper provesthe Theorem 1 with the help of Dyson’s rank conjectures N(0,5,5n +1), N(2,5, 5n +1) and proves the Theorem 2 with thehelp of Garvan’s theta series and Dyson’s rank conjectures N(1,5, 5n+2), N(2,5, 5n+2), respectively. An attempt has beentaken here to the development of the Ramanujan’s works with the contributions of Dyson and Garvan. Definitions andsimple mathematical calculations are presented here to make the paper easier to the common readers.