Formulas of Ramanujan for the power series coefficients of certain quotients of Eisenstein series

Abstract

In their last published paper [9], [16, pp. 310–321], G. H. Hardy and S. Ramanujan derived infinite series representations for the coefficients of certain modular forms of negative weight which are not analytic in the upper half-plane. In particular, they examined in detail the coefficients of the reciprocal of the Eisenstein series E6(τ). While confined to the sanitarium, Matlock House, in 1918, Ramanujan wrote several letters to Hardy about the coefficients in the power series expansions of certain quotients of Eisenstein series. These letters are photocopied in [18, pp. 97–126], and printed versions with commentary can be found in [6, pp. 175–191]. In these letters, Ramanujan recorded formulas for the coefficients of several quotients of Eisenstein series not examined by Hardy and him in [9]. These claims fall into two related classes. In the first class are formulas for coefficients that arise from the main theorem of Hardy and Ramanujan, or a slight modification of it, and these results have been proved in a paper by Berndt and Bialek [5]. Those in the second class, which we prove in this paper, are much harder to prove. To establish the first main result, we need an extension of Hardy and Ramanujan’s theorem due to H. Petersson [11]. To prove the second primary result, we need to first extend work of H. Poincare [14], Petersson [11], [12], [13], and J. Lehner [10] to cover double poles. In all cases, the formulas have a completely different shape from those arising from modular forms analytic in the upper half-plane, such as the famous infinite series for the partition function p(n) arising from the reciprocal of the Dedekind eta-function. As we shall see in the sequel, the series examined in this paper are very rapidly convergent, even more so than those arising from modular forms analytic in the upper half-plane, so that truncating a series, even with a small number of terms, provides a remarkable approximation. Using Mathematica, we calculated several coefficients and series approximations for the two primary functions 1/B(q) and 1/B(q) (defined below) examined by Ramanujan. As will be seen from the first table, the coefficient of q in 1/B(q), for example, has 17 digits, while just two terms of Ramanujan’s infinite series representation calculate this coefficient with