A Class of the Generalized Ramanujan Tau Numbers and Their Associated Partition Functions

In this paper, the authors derive some believed-to-be new recursion and explicit formulas for the generalized Ramanujan numbers τs(n)(s∈N∪{−1}), where, as usual, N is the set of positive integers. The authors consider the associated partition functions and derive connections of the Eisenstein series with the numbers τs(n). Several related corollaries and consequences of each of the presented results are also given. The paper concludes by presenting an open problem that is related to one of these results.