Mixed Berndt-Type Integrals and Generalized Barnes Multiple Zeta Functions

TL;DR

This paper introduces mixed Berndt-type integrals involving hyperbolic functions, derives closed-form expressions using elliptic functions and gamma constants, and explores their connection to generalized Barnes multiple zeta functions.

Contribution

It defines and evaluates new mixed Berndt-type integrals and establishes their relation to generalized Barnes multiple zeta functions, providing explicit formulas and examples.

Findings

Closed-form expressions for mixed Berndt-type integrals.

Connection between integrals and generalized Barnes zeta functions.

Explicit evaluations involving gamma constants and zeta functions.

Abstract

In this paper, we define and study four families of Berndt-type integrals, called mixed Berndt-type integrals, which contain (hyperbolic) sine and cosine functions in the integrand function. Using contour integration, these integrals are first converted to some hyperbolic (infinite) sums of Ramanujan type, all of which can be calculated in closed form by comparing both the Fourier series expansions and the Maclaurin series expansions of certain Jacobi elliptic functions. These sums can be expressed as rational polynomials of $\Gamma(1/4)$ and $\pi^{-1}$ which give rise to the closed formulas of the mixed Berndt-type integrals we are interested in. Moreover, we also present some interesting consequences and illustrative examples. Additionally, we define a generalized Barnes multiple zeta function, and find a classic integral representation of the generalized Barnes multiple zeta function. Furthermore, we give an alternative evaluation of the mixed Berndt-type integrals in terms of the generalized Barnes multiple zeta function. Finally, we obtain some direct evaluations of rational linear combinations of the generalized Barnes multiple zeta function.