Eisenstein series and theta functions to the septic base |
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Authors: | Heng Huat Chan |
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Institution: | a National University of Singapore, Department of Mathematics, 2 Science Drive 2, Singapore 117543, Singapore b Massey University, Albany Campus, Private Bag 102 904, North Shore Mail Centre, Auckland, New Zealand |
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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 and the eta quotient η4(7τ)/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. |
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Keywords: | 11F11 11F20 14K25 33D52 33E05 |
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