RAMANUJAN’S SPT-CRANK FOR MARKED OVERPARTITIONS

Abstract

In 1916, Ramanujan’s showed the spt-crank for marked overpartitions. The corresponding special functions ,  and are found in Ramanujan’s notebooks, part 111. In 2009, Bingmann, Lovejoy and Osburn defined the generating functions for , and . In 2012, Andrews, Garvan, and Liang  defined the  in terms of partition pairs. In this article the number of smallest parts in the overpartitions of n with smallest part not overlined, not overlined and odd, not overlined and even are   discussed, and the vector partitions and - partitions with 4 components, each a partition with certain restrictions are also discussed. The generating functions  ,  , , ,  are shown with the corresponding results in terms of modulo 3, where the generating functions  ,  are collected from Ramanujan’s notebooks, part 111.  This paper shows how to prove the Theorem 1 in terms of ,Theorem 2 in terms of  and Theorem 3 in terms of  respectively with the numerical examples, and shows how to prove the Theorems 4,5 and 6 with the help of  in terms of partition pairs. In 2014, Garvan and Jennings-Shaffer  are able to defined the  for marked overpartitions. This paper also shows another results with the help of 6 -partition pairs of 3, help of 20 -partition pairs of 5 and help of 15 -partition pairs of 8 respectively.