INTEGRAL REPRESENTATIONS OF EULER-TYPE FOR THE QUADRUPLE HYPERGEOMETRIC FUNCTIONS $F^{(4)}_{20}$

The authors establish a set of fifteen new integral representations of Euler-type for the Sharma and Parihar hypergeometric function in four variables $F^{(4)}_{20}$; whose kernels include the quadruple functions $K_{1}, K_{10}, F^{(4)}_{14}$ and $X^{(4)}_{8}$; the Exton hypergeometric functions of three variables $X_{4}$; the Lauricella functions of three variable $F_{E}, F_{N}$ and $F_{R}$; the Appell’s series of two variables $F_{2}, F_{3}$ and $F_{4}$; and the Gauss hypergeometric function.