A Partition Function with Some Congruence Condition

Abstract

Introduction. It is well known that the number p(n) of unrestricted partitions of a positive integer n can be expressed by a convergent series (see [9]). In this paper we shall be concerned with the number p (n; a, M) of partitions of n into positive summands congruent to ? a modulo Mi, where a, M are integers with M ? 2. This partition function p (n; a, M) has been treated for certain special values of M; namely, M 2 by Hua [2], M==5 by Lehner [5], M=6 by Niven [8], and M = p (p a prime > 3) by Livingood [6]; a convergent series representation of p (n; a, M) being obtained in each case. The main object of the present paper is to derive a convergent series for p (n; a, M) in which M assumes general values. We may suppose without loss of generality that 0 1 and d I n, it reduces to p (n/d; a/d, M/d). Further we remark that the case M =2 (partitions into odd summands) is equivalent to the case M = 4, i. e., p (n; 1, 2) -p (n; 1, 4). Consequently it suffices to consider the case M ? 3. We apply the Hardy-Ramanujan method with modifications due to Rademacher [10]. In order to utilize this method, however, it is necessary to solve the following two subsidiary problems: The first problem is to find a suitable transformation equation for the generating function of p (n; a, M). The usual method of contour integration seems to be rather complicated in our case. However, we can show, in a simple way, the existence of the transformation equation, the proof of which being based on a certain functional equation recently obtained by the author [3]. This will be done in Section 1 of this paper. The second is to get a non-trivial estimate for a certain exponential sum which is introduced as a consequence of the transformation equation. We shall