Evaluation of reciprocal sums of hyperbolic functions using quasimodular forms

Abstract

This paper studies eight families of infinite series involving hyperbolic functions. Under some conditions, these series are linear combinations of derivatives of Eisenstein series. The paper gives a systematic method for computing the values of these series at CM points. The approach utilizes complex multiplication theory, the structure of the rings of modular forms and quasimodular forms, and certain differential operators defined on these rings. This paper also expresses the generalized reciprocal sums of Fibonacci numbers as the special values of the series mentioned above. Thus it gives some algebraic independence results about the generalized reciprocal sums of Fibonacci numbers.