Eisenstein series and theta functions to the septic base

Abstract

We develop a theory for Eisenstein series to the septic base, which was started by S. Ramanujan in his "Lost Notebook." We show that two types of septic Eisenstein series may be parameterized in terms of the septic theta function ∑m = - ∞ ∞ ∑n = - ∞ ∞ qm2 + m n + 2 n2 and the eta quotient η4 (7 τ) / η4 (τ). This is accomplished by constructing elliptic functions which have the septic Eisenstein series as Taylor coefficients. The elliptic functions are shown to be solutions of a differential equation, and this leads to a recurrence relation for the septic Eisenstein series. © 2007 Elsevier Inc. All rights reserved.