Evaluation of the symmetrized Mordell-Tornheim zeta function

TL;DR

This paper evaluates the symmetrized Mordell-Tornheim zeta function, revealing simple representations in terms of Bell polynomials and special functions, and providing explicit values for small depths.

Contribution

It introduces explicit formulas for the symmetrized Mordell-Tornheim zeta function at equal weights, connecting it to Bell polynomials and special functions, and extends understanding beyond classical versions.

Findings

Explicit formulas for (1,...,1) in terms of Bell polynomials.

Representation of the zeta function using derivatives of a special logarithmic function.

Explicit values provided for depths 1 to 10.

Abstract

In this paper we evaluate the symmetrized Mordell-Tornheim zeta function defined as \begin{equation*} \overline{\zeta}_n(w_1, \ldots, w_n) = \sum_{\substack{a_1, \ldots, a_n \in \mathbb{Z}^* \\ a_1 + \ldots + a_n = 0}} \frac{1}{\left| a_1^{w_1} \cdots a_n^{w_n} \right|} \end{equation*} where $n \ge 1$ is a positive integer representing the depth and $w_1, \ldots, w_n \ge 1$ are positive integers representing the weight $w = w_1 + \ldots + w_n$ of the function. Compared to the classical Mordell-Tornheim zeta function $\zeta_{MT,n}(w_1, \ldots, w_n; w_{n+1})$ which is restricted to the positive orthant (hyperoctant), the symmetrized one spans the entire $(n-1)$-dimensional hyperplane. We show that when the depth and the weight of the function are equal, that is for $\overline{\zeta}_n(1, \ldots, 1)$, it has a remarkably simple representation in terms of standard functions: \begin{equation*} \overline{\zeta}_n(1, \ldots, 1) = B_n(f^{(1)}(0), \ldots, f^{(n)}(0)) \end{equation*} where $B_n$ is $n$-th complete exponential Bell polynomial and $f^{(n)}(0)$ is $n$-th derivative at $x=0$ of function $f(x)$ defined as: \begin{equation*} f(x) = \ln \binom{-2x}{-x} \end{equation*} Additionally, we show the value can be expressed using the following polynomials with positive integer coefficients over the values of zeta function: \begin{equation*} \overline{\zeta}_n(1, \ldots, 1) = B_n(0, (2^2 - 2) \Gamma(2) \zeta(2), \ldots, (2^n - 2) \Gamma(n) \zeta(n)) \end{equation*} or equivalently, over the values of eta function: \begin{equation*} \overline{\zeta}_n(1, \ldots, 1) = B_n(0, 2^2 \Gamma(2) \eta(2), \ldots, 2^n \Gamma(n) \eta(n)) \end{equation*} The list of explicit values for small $1 \le n \le 10$ is available in the appendix.