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Berndt-type Integrals: Unveiling Connections with Barnes Zeta and Jacobi Elliptic Functions

Zachary P. Bradshaw, Christophe Vignat

TL;DR

This work analyzes Berndt-type integrals $I_{\pm}(s,p)$ through the Barnes zeta function framework, extending Xu and Zhao by obtaining Barnes zeta representations and analytic continuations for these integrals. It then leverages Kuznetsov’s results to connect Berndt-type integrals with Jacobi elliptic functions, enabling expressions in terms of moment polynomials of Lomont and Brillhart and deriving Lambert-series and Eisenstein-series representations. The paper also extends the analysis to the difference case, introduces generating functions, probabilistic interpretations, and lattice-sum expansions, and derives arithmetical results modulo 10 and 3 for these integrals in special elliptic regimes. Collectively, the results unify multiple strands—Barnes zeta, elliptic functions, Lambert/Eisenstein series, and modular polynomials—revealing deeper structural insights and potential Dirichlet analogs in the theory of Berndt-type integrals.

Abstract

We address a class of definite integrals known as Berndt-type integrals, highlighting their role as specialized instances within the integral representation framework of the Barnes-zeta function. Building upon the foundational insights of Xu and Zhao, who adeptly evaluate these integrals using rational linear combinations of Lambert-type series and derive closed-form expressions involving products of $Γ^4(1/4)$ and $π^{-1}$, we uncover direct evaluations of the Barnes-zeta function. Moreover, our inquiry leads us to establish connections between Berndt-type integrals and Jacobi elliptic functions, as well as moment polynomials investigated by Lomont and Brillhart, a relationship elucidated through the seminal contributions of Kuznetsov. In this manner, we extend and integrate these diverse mathematical threads, unveiling deeper insights into the intrinsic connections and broader implications of Berndt-type integrals in special function and integration theory.

Berndt-type Integrals: Unveiling Connections with Barnes Zeta and Jacobi Elliptic Functions

TL;DR

This work analyzes Berndt-type integrals through the Barnes zeta function framework, extending Xu and Zhao by obtaining Barnes zeta representations and analytic continuations for these integrals. It then leverages Kuznetsov’s results to connect Berndt-type integrals with Jacobi elliptic functions, enabling expressions in terms of moment polynomials of Lomont and Brillhart and deriving Lambert-series and Eisenstein-series representations. The paper also extends the analysis to the difference case, introduces generating functions, probabilistic interpretations, and lattice-sum expansions, and derives arithmetical results modulo 10 and 3 for these integrals in special elliptic regimes. Collectively, the results unify multiple strands—Barnes zeta, elliptic functions, Lambert/Eisenstein series, and modular polynomials—revealing deeper structural insights and potential Dirichlet analogs in the theory of Berndt-type integrals.

Abstract

We address a class of definite integrals known as Berndt-type integrals, highlighting their role as specialized instances within the integral representation framework of the Barnes-zeta function. Building upon the foundational insights of Xu and Zhao, who adeptly evaluate these integrals using rational linear combinations of Lambert-type series and derive closed-form expressions involving products of and , we uncover direct evaluations of the Barnes-zeta function. Moreover, our inquiry leads us to establish connections between Berndt-type integrals and Jacobi elliptic functions, as well as moment polynomials investigated by Lomont and Brillhart, a relationship elucidated through the seminal contributions of Kuznetsov. In this manner, we extend and integrate these diverse mathematical threads, unveiling deeper insights into the intrinsic connections and broader implications of Berndt-type integrals in special function and integration theory.
Paper Structure (20 sections, 28 theorems, 248 equations, 1 table)

This paper contains 20 sections, 28 theorems, 248 equations, 1 table.

Table of Contents

  1. Introduction
  2. Connections with the Barnes Zeta Function
  3. An integral representation
  4. Laplace transforms
  5. Analytic continuation
  6. Kuznetsov's Approach
  7. Connection to Berndt Integrals
  8. Moment Polynomials of Lomont and Brillhart
  9. Lambert Series Representation
  10. Eisenstein Series Representation
  11. Extension of Kuznetsov's Result: The Difference Case
  12. Generating Function
  13. The Polynomials of Lomont and Brillhart
  14. A Probabilistic Approach
  15. Lambert Series Representation
  16. ...and 5 more sections

Key Result

Proposition 1

Let $\Re(s)>N$, $\Re(w)>0$, and $\Re(a_j)>0$ for $j=1,\ldots,N$. Then The alternating version is

Theorems & Definitions (51)

  • Proposition 1
  • proof
  • Corollary 1
  • Proposition 2
  • proof
  • Remark 1
  • Corollary 2
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 41 more