Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Notes on certain binomial harmonic sums of Sun's type

Yajun Zhou

TL;DR

This work bridges Sun's binomial-harmonic sums with automorphic and modular structures by embedding Legendre functions and their products into moduli-space and CM frameworks. It develops a unified program based on the Frobenius–Zagier process, Clausen couplings, Green's functions, and Bailey–Meijer techniques to express and evaluate a wide class of Sun-type series in closed form, including Ramanujan–Sun and Epstein zeta contexts. Key contributions include modular parametrizations of Legendre–Sun series at CM points, new derivative/integral identities for Legendre functions, and automorphic representations of Green's and Epstein zeta functions that yield explicit constants and CM values. The results illuminate deep connections between hypergeometric functions, modular forms, and binomial-harmonic sums, with implications for closed-form evaluations, special values of gamma functions, and related supercongruence phenomena.

Abstract

We prove and generalize some recent conjectures of Z.-W. Sun on infinite series whose summands involve products of harmonic numbers and several binomial coefficients. We evaluate various classes of infinite sums in closed form by interpreting them as automorphic objects on the moduli spaces for Legendre curves $Y^{ g+1}=(1-X)^{ g}X(1-t X)$ of positive genera $ g\in\{1,2,3,5\}$.

Notes on certain binomial harmonic sums of Sun's type

TL;DR

This work bridges Sun's binomial-harmonic sums with automorphic and modular structures by embedding Legendre functions and their products into moduli-space and CM frameworks. It develops a unified program based on the Frobenius–Zagier process, Clausen couplings, Green's functions, and Bailey–Meijer techniques to express and evaluate a wide class of Sun-type series in closed form, including Ramanujan–Sun and Epstein zeta contexts. Key contributions include modular parametrizations of Legendre–Sun series at CM points, new derivative/integral identities for Legendre functions, and automorphic representations of Green's and Epstein zeta functions that yield explicit constants and CM values. The results illuminate deep connections between hypergeometric functions, modular forms, and binomial-harmonic sums, with implications for closed-form evaluations, special values of gamma functions, and related supercongruence phenomena.

Abstract

We prove and generalize some recent conjectures of Z.-W. Sun on infinite series whose summands involve products of harmonic numbers and several binomial coefficients. We evaluate various classes of infinite sums in closed form by interpreting them as automorphic objects on the moduli spaces for Legendre curves of positive genera .

Paper Structure

This paper contains 21 sections, 22 theorems, 147 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Legendre functions and Sun's series
  3. Frobenius--Zagier process and Legendre--Sun series
  4. Modular parametrizations of Legendre--Sun series
  5. Derivatives and integrals related to Legendre--Sun series
  6. Derivatives with respect to degrees of Legendre functions
  7. Integrals involving Legendre--Sun series
  8. Clausen couplings and Ramanujan--Sun series
  9. Clausen couplings of Legendre functions
  10. Ramanujan--Sun series
  11. Some series descending from Clausen couplings
  12. Derivatives and differential equations for products of two Legendre functions
  13. Integrals over products of two Legendre functions
  14. Green--Epstein variations on Sun's series
  15. Derivatives of Legendre functions with respect to their degrees
  16. ...and 6 more sections

Key Result

Theorem 2.1

For $t\in[-1,0)$, one has $\widetilde{\mathsf S}_{\nu}^L(t)=\lim_{\varepsilon\to0^+}\widetilde{\mathsf S}_{\nu}^L(t\pm i\varepsilon^+)$ for $\nu\in\left\{-\frac{1}{2},-\frac{1}{3},-\frac{1}{4},-\frac{1}{6}\right\}$.$\blacksquare$

Theorems & Definitions (37)

  • Theorem 2.1: Legendre--Sun series
  • Lemma 2.2: Frobenius--Zagier
  • proof
  • Corollary 2.3: Some arithmetic properties of Legendre--Sun series
  • proof
  • Corollary 2.4: Binomial modulations of Legendre--Sun series
  • proof
  • Corollary 2.5: Harmonic modulations of Legendre--Sun series
  • proof
  • Theorem 3.1: Ramanujan--Sun series
  • ...and 27 more