Partitions into Odd and Unequal Parts

Abstract

and asymptotic formulas for q(n), the number of partitions of a positive integer n into positive odd summands. The necessary transformation equations as well as estimates of the magnitude of certain sums of roots of unity were obtained by using essentially the same procedures as Lehner [4], while the circle dissection method of Rademacher [6] was employed for the integration. In the present paper we shall impose an additional restriction on the partitions, namely that the summands be distinct. That is, we wish to find a convergent series and asymptotic formulas for Q(n), the number of partitions of a positive integer n into odd and unequal parts. Q (n) also represents the number of self-conjugate partitions of n (see pp. 278-9 in [3]). The investigation is parallel to that in [2], and free use will be made of the results obtained in the earlier paper whenever they are applicable. 2. The transformation equations. Several generating fuLnctions, each convergent in the interior of the unit circle, will be needed. We list these here. 00 00