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Series involving central binomial coefficients and harmonic numbers of order 2

Zhi-Wei Sun, Yajun Zhou

TL;DR

The paper develops a modular-parametrization framework for infinite series whose terms involve central binomial coefficients and second-order harmonic numbers, enabling reductions to special values of Dirichlet L-functions for certain convergence rates. By building a toolkit around Dedekind-eta, modular lambda, and E4, and pairing Epstein zeta functions with Eichler integrals, it translates challenging hypergeometric-harmonic sums into Dirichlet L-value evaluations. Hypergeometric deformations and variation-of-parameters yield integral representations tied to the complete elliptic integral K, which, when coupled with Lambert-Ramanujan identities and Eichler sum rules, permit explicit reductions at special quadratic points. These methods culminate in proving Sun's conjectured identities, including those featuring L_{-7}(2), and demonstrate a principled route from modular parametrizations to concrete L-value evaluations with potential applicability to related Ramanujan-type sums. The work thus highlights the deep connections among hypergeometric series, modular forms, Eichler integrals, Epstein zeta functions, and Dirichlet L-values, with implications for exact evaluations of a broad class of series.

Abstract

We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and second-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to reduce the corresponding series to special values of the Dirichlet $L$-values. For example, we establish the following identity that has been recently conjectured by Sun:\[\sum_{k=0}^\infty\binom{2k}{k}^3\left[ \mathsf H_{2k}^{(2)}-\frac{25}{92}\mathsf H_{ k}^{(2)} +\frac{735L_{-7}(2)-86π^{2}}{1104}\right]\frac{1}{4096^{k}}=0,\] where $ \mathsf H^{(2)}_k:= \sum_{0<j\leq k}\frac{1}{j^2}$ and $ L_{-7}(2):= \sum_{n=1}^\infty\left(\frac{-7}{n}\right)\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{4^{2}}-\frac{1}{5^{2}}-\frac{1}{6^{2}}+\frac{1}{8^{2}}+\cdots $.

Series involving central binomial coefficients and harmonic numbers of order 2

TL;DR

The paper develops a modular-parametrization framework for infinite series whose terms involve central binomial coefficients and second-order harmonic numbers, enabling reductions to special values of Dirichlet L-functions for certain convergence rates. By building a toolkit around Dedekind-eta, modular lambda, and E4, and pairing Epstein zeta functions with Eichler integrals, it translates challenging hypergeometric-harmonic sums into Dirichlet L-value evaluations. Hypergeometric deformations and variation-of-parameters yield integral representations tied to the complete elliptic integral K, which, when coupled with Lambert-Ramanujan identities and Eichler sum rules, permit explicit reductions at special quadratic points. These methods culminate in proving Sun's conjectured identities, including those featuring L_{-7}(2), and demonstrate a principled route from modular parametrizations to concrete L-value evaluations with potential applicability to related Ramanujan-type sums. The work thus highlights the deep connections among hypergeometric series, modular forms, Eichler integrals, Epstein zeta functions, and Dirichlet L-values, with implications for exact evaluations of a broad class of series.

Abstract

We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and second-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to reduce the corresponding series to special values of the Dirichlet -values. For example, we establish the following identity that has been recently conjectured by Sun:\[\sum_{k=0}^\infty\binom{2k}{k}^3\left[ \mathsf H_{2k}^{(2)}-\frac{25}{92}\mathsf H_{ k}^{(2)} +\frac{735L_{-7}(2)-86π^{2}}{1104}\right]\frac{1}{4096^{k}}=0,\] where and .
Paper Structure (5 sections, 4 theorems, 57 equations, 1 table)

This paper contains 5 sections, 4 theorems, 57 equations, 1 table.

Table of Contents

  1. Introduction
  2. Infinite series involving $\binom{2k}k^3$, $\mathsf H_k^{(2)}$, and $\mathsf H_{2k}^{(2)}$
  3. Hypergeometric deformations and variation of parameters
  4. Lambert--Ramanujan series, Eichler sum rules, and Epstein zeta functions
  5. Eichler sum rules

Key Result

Theorem 1.1

If we have either $2z/i\geq1$, or $\mathop{\mathrm{Re}}\nolimits z=\frac{1}{2}$ and $\mathop{\mathrm{Im}}\nolimits z\geq\frac{1}{\sqrt{2}}$, then

Theorems & Definitions (8)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:H2']}
  • Proposition 2.3
  • proof