  The multiple gamma function $\Gamma_n$, defined by a recurrence-functional
equation as a generalization of the Euler gamma function, was originally
introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the
pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma
function has been revived. This paper discusses some theoretical aspects of the
$\Gamma_n$ function and their applications to summation of series and infinite
products.
