A Partition Function with the Prime Modulus P > 3

Abstract

1. Recently Lehner [2]1 developed a convergent series for pi(n) and p2(n), the number of partitions of a positive integer n into summands of the form 51 ? 1 and 51 + 2 respectively. The purpose of this paper is to develop a similar series for p1(n), p2 (n), * (,-1)/2(n), the number of partitions of a positive integer n into summands ofthe form pl ? 1, pl ? 2, pl ? (p 1)72, p being a prime greater than 3. Here, as in the previous paper, we follow the method of Rademacher [5]. We consider the generating functions