Multiple Euler integral representations for the Kampé de Fériet functions

By the aid of Appell, Humbert and Bessel functions, the integral representations for a Kampé de Fériet function are found. The validity of integral representations for a Kampé de Fériet function of general form are proved. Conditions, under which these representations are expressed in terms of products of two generalized hypergeometric functions are found. Examples, in which the integral representation of the Kampé de Fériet function containing Appell, Humbert or Bessel functions, are identified.