ON GENERALIGATIONS OF PARTITION FUNCTIONS

In 1742, Leonhard Euler invented the generating function for P(n). Godfrey Harold Hardy said Srinivasa Ramanujan was the first, and up to now the only, Mathematician to discover any such properties of P(n). In 1916, Ramanujan defined the generating functions for X(n),Y(n) . In 2014, Sabuj developed the generating functions for . In 2005, George E. Andrews found the generating functions for In 1916, Ramanujan showed the generating functions for , , and . This article shows how to prove the Theorems with the help of various auxiliary functions collected from Ramanujan's Lost Notebook. In 1967, George E. Andrews defined the generating functions for P1r (n) and P2r (n). In this article these generating functions are discussed elaborately. This article shows how to prove the theorem P2r (n) = P3r (n) with a numerical example when n = 9 and r = 2. In 1995, Fokkink, Fokkink and Wang defined the in terms of , where is the smallest part of partition . In 2013, Andrews, Garvan and Liang extended the FFW-function and defined the generating function for FFW (z, n) in differnt way.