DEVELOPMENT OF LOVEJOY AND OSBURN'S OVERPARTITION FUNCTIONS

In 2008, Lovejoy and Osburn defined the generating function for Pn .In 2009, Byungchan Kim defined the generating function for P n 2 .This paper shows how to discuss the generating functions for Pn and P n 2 . Byungchan Kim also defined P n k with increasing relation and overpartition congruences mod 4,8 and 64. In 2006, Berndt found the relation ( ) ( ) d1,4 n d3,4 n has two values with certain restrictions and various formulae by the common term (n) .This paper shows how to prove the four Theorems about overpartitions modulo 8.These Theorems satisfy the arithmetic properties of the overpartition function modulo 8