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Evaluation of reciprocal sums of hyperbolic functions using quasimodular forms

Wei Wang

TL;DR

The paper develops a systematic framework to evaluate eight families of hyperbolic-sum series by representing them as linear combinations of derivatives of Eisenstein series within the theory of quasimodular forms. It leverages CM theory, modular polynomials, and Maass-type differentiation to compute CM-values and higher derivatives, yielding explicit evaluations and a main classification theorem for when these series arise as Eisenstein-derivatives. By connecting these series to generalized Fibonacci zeta functions, the authors obtain algebraic-independence results for special values, demonstrating that many Fibonacci-related sums are transcendental and independent under precise parity constraints. The work provides a cohesive approach that unifies hyperbolic reciprocal sums, modular forms, and Fibonacci zeta values, offering a practical toolkit for both exact CM evaluations and transcendence-type conclusions with potential arithmetic applications.

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.

Evaluation of reciprocal sums of hyperbolic functions using quasimodular forms

TL;DR

The paper develops a systematic framework to evaluate eight families of hyperbolic-sum series by representing them as linear combinations of derivatives of Eisenstein series within the theory of quasimodular forms. It leverages CM theory, modular polynomials, and Maass-type differentiation to compute CM-values and higher derivatives, yielding explicit evaluations and a main classification theorem for when these series arise as Eisenstein-derivatives. By connecting these series to generalized Fibonacci zeta functions, the authors obtain algebraic-independence results for special values, demonstrating that many Fibonacci-related sums are transcendental and independent under precise parity constraints. The work provides a cohesive approach that unifies hyperbolic reciprocal sums, modular forms, and Fibonacci zeta values, offering a practical toolkit for both exact CM evaluations and transcendence-type conclusions with potential arithmetic applications.

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.
Paper Structure (12 sections, 16 theorems, 170 equations)

This paper contains 12 sections, 16 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. expressing series as derivatives of Eisenstein series
  4. CM evaluation of modular forms and quasimodular forms
  5. The general methods
  6. Computation: CM values of modular forms
  7. Computation: higher-order derivatives of modular forms
  8. Explicit examples
  9. reciprocal sums of Fibonacci numbers
  10. Generalized Fibonacci zeta function
  11. Results on algebraic independence
  12. Examples

Key Result

Proposition 3.1

The $q$-series in the first class can be expressed in terms of $A_s(q)$: The $q$-series in the second class can be expressed in terms of $G_s(q)$ and $N_s(q)$:

Theorems & Definitions (27)

  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • proof
  • ...and 17 more