A Definite Integral of a Product of Two Polylogarithms

Using the product theorem for the Mellin transform, a definite integral depending on several parameters and containing a product of two polylogarithm functions (or two logarithms in the degenerate case) is replaced by a Mellin–Barnes integral, which in turn is evaluated by residue techniques. The results are given in terms of infinite series of hypergeometric type. Some special cases for which the infinite series can be expressed in closed form are also considered.