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

TL;DR

This paper explores Berndt-type integrals, revealing their connections with Barnes zeta functions, Jacobi elliptic functions, and moment polynomials, thereby deepening understanding of their role in special function theory.

Contribution

It establishes new links between Berndt-type integrals, Barnes zeta functions, and elliptic functions, extending prior evaluations and unifying diverse mathematical concepts.

Findings

Direct evaluations of Barnes-zeta function related to Berndt-type integrals.

Connections established between Berndt-type integrals and Jacobi elliptic functions.

Extended understanding of the role of Berndt-type integrals in special functions.

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 $\Gamma^4(1/4)$ and $\pi^{-1}$, 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.