Some properties of partitions. II

have given a complete account of Dyson's rank functions for the cases g = 5 and 2 = 7.We shall adopt throughout the notation of [7] together with some of our own, and require to refer so constantly to [7] that we have decided to regard this paper as a sequel, numbering the sections, theorems, and equations, consecutively.Thus any reference in this paper to § §1 to 6, Theorems 1 to 5, and Lemmas 1 to 8, is to the result so'numbered in [7].Our main object in this paper is to carry out the corresponding investigation for q = 11, in particular to obtain the values of the rbc(d) for q = 11.These values are considerably more complicated than those for q = 5 and q = 7.Their interest lies largely in the methods by which they were obtained; as in [7] the complete set of values had to be obtained empirically before the proof of the correctness of any one was possible.We shall describe these methods in §9.§8 contains some preliminary results for the functions P(a) with 2 = 11, and in §10 we state and prove the main results in Theorem 6. §11 contains the evaluation of zZm-o ynp(lln-\-6) in a form similar to Ramanujan's results for g = 5 and q -1, although less elegant.In the table of notation and list of references at the end we only give the notation and references introduced in this paper and not that of [7].We observe finally that some letters occur twice in the text in different senses, but the contexts are so different as to give no danger of confusion.