DEVELOPMENT OF FOKKINK-FOKKINK-WANG'S GENERATING FUNCTION FOR FFW(n)

In 1995, R. Fokkink, W. Fokkink and Wang defined the FFW(n)in terms of s(pi) , where s(pi) is the smallest part of partition? . In 2008, Andrews obtained the generating function for FFW(n) . In 2013, Andrews, Garvan and Liang extended the FFW-function and obtained the similar expressions for the spt-function and then defined the spt-crank generating functions. They also defined the generating function for FFW(z,n)in various ways. This paper shows how to find the number of partitions of n into distinct parts with certain conditions and shows how to prove the Theorem 1 by induction method. This paper shows how to prove the Theorem 2 with the help of two generating functions.