Mixed Berndt-Type Integrals and Generalized Barnes Multiple Zeta Functions
Jianing Zhou
TL;DR
The paper develops a framework for mixed Berndt-type integrals containing $(\\sinh x)$ and $\sin x$, converting them via contour integration into Ramanujan-type hyperbolic sums that admit closed forms in terms of $\\Gamma(1/4)$ and $\\pi^{-1}$. By leveraging Fourier and Maclaurin expansions of Jacobi elliptic functions, it derives explicit evaluations and structural relations, complemented by detailed corollaries and examples. A generalized Barnes multiple zeta function is introduced with a classical integral representation, and the work establishes connections between these mixed integrals and Barnes zeta values, yielding direct evaluations of rational linear combinations of the zeta functions. Overall, the results unify Berndt-type integrals with Barnes zeta theory and expand the toolkit for exact evaluations in this domain.
Abstract
In this paper, we define and study four families of Berndt-type integrals, called mixed Berndt-type integrals, which contains (hyperbolic) sine and cosine functions in the integrand function. By contour integration, these integrals are first converted to some hyperbolic (infinite) sums of Ramanujan type, all of which can be calculated in closed forms by comparing both the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions. These sums can be expressed as rational polynomials of Γ(1/4) and π^{-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. Further, 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.