Euler-type integral representations for the Kampé de Fériet functions

In this paper, the Kampé de Fériet functions of arbitrary orders and their Euler-type integral representations are studied. The general form of the integral representations for a Kampé de Fériet function are proved. Conditions, under which these representations are expressed in terms of products of two gen-eralized hypergeometric functions, are found. Examples are identified in which the integral representation of the Kampé de Fériet function contains an elementary function or a known second-order hypergeometric function of two variables.