On a certain convolution of polylogarithms

In this paper, we consider certain double series analogously to Tornheim's double series and real analytic Eisenstein series. By computing double integrals in two kinds of ways, we give some functional relations between these double series and the ordinary polylogarithms. The technique to prove them is a certain generalization of the known one given by Prudnikov and Kanemitsu-Tanigawa-Yoshimoto. As concrete examples, these functional relations give new evaluation formulas for certain double series expressed by the Riemann zeta values, and some analogues of Goncharov's formulas for double polylogarithms. 10.1017/S000497271100339