Development of Partition Functions of Ramanujan’s Works

In 1986, Dyson defined the rank of a partition as the largest part of a partition minus the number of parts of . In 1988, 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 proves the 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 the help of Garvan’s theta series and Dyson’s rank conjectures N(1,5, 5n+2), N(2,5, 5n+2), respectively. An attempt has been taken here to the development of the Ramanujan’s works with the contributions of Dyson and Garvan. Definitions and simple mathematical calculations are presented here to make the paper easier to the common readers.