For a given positive integer \(r\), the Mordell-Tornheim multiple zeta function is defined by \[ \zeta_{MT,r}(s_1,\ldots, s_r;s_{r+1})=\sum_{n_1, n_2, \ldots, n_r \in \mathbb{N}}\frac{1}{n_1^{s_1}n_2^{s_2}\cdots n_r^{s_r}(n_1+\cdots +n_r)^{s_{r+1}}} \] on the subset \(V_{r+1}=\lbrace (s_1,\ldots,s_r,s_{r+1}); \operatorname{Re}(s_{r+1})+ \sum_{\ell=1}^j \operatorname{Re}(s_{k_\ell})>j\) with \( 1 \leq k_1 < \cdots < k_j \leq r\) for any \(j = 1,\ldots, r \rbrace\) of \(\mathbb{C}^{r+1}\). The function \(\zeta_{MT,r}\) is holomorphic on \(V_{r+1}\) and it can be meromorphically continued to the whole \(\mathbb{C}^{r+1}\). By showing a relation between the Mordell-Tornheim multiple zeta-function \(\zeta_{MT,r}\) and the confluent hypergeometric function \[ \Psi(a,c;x)=\frac{1}{\Gamma(a)}\int_{0}^{ e^{i\phi}\infty}e^{-xy}y^{a-1}(1+y)^{c-a-1}\, dy, \] where \(\operatorname{Re} (a) >0,\, -\pi <\phi < \pi,\, | \phi + \arg x| < \frac{\pi}{2}\), T. Okamoto and T. Onozuka established the functional equation for the Mordell-Tornheim multiple zeta-function \(\zeta_{MT,r}\) [Funct. Approximatio, Comment. Math. 55, No. 2, 227–241 (2016; Zbl 1406.11086)].
The aim of this note is to give an alternative and relatively simple proof of the functional equation for the Mordell-Tornheim multiple zeta-function \(\zeta_{MT,r}\) based on the prior work of F. V. Atkinson [Acta Math. 81, 353–376 (1949; Zbl 0036.18603)] for double zeta function during the study of the mean-value of the Riemann zeta function.