Honam Mathematical J. 32 (2010), No. 3, pp. 389–397 CERTAIN INTEGRAL REPRESENTATIONS OF EULER TYPE FOR THE EXTON FUNCTION X 5 Junesang Choi, Anvar Hasanov and Mamasali Turaev Abstract. Exton introduced 20 distinct triple hypergeometric func- tions whose names are X i (i =1, ..., 20) to investigate their twenty Laplace integral representations whose kernels include the conflu- ent hypergeometric functions 0 F 1 , 1 F 1 , a Humbert function Ψ 2 ,a Humbert function Φ 2 . The object of this paper is to present 25 (pre- sumably new) integral representations of Euler types for the Exton hypergeometric function X 5 among his twenty X i (i =1, ..., 20), whose kernels include the Exton function X5 itself, the Exton func- tion X 6 , the Horn’s functions H 3 and H 4 , and the hypergeometric function F = 2 F 1 . 1. Introduction Exton [4] introduced 20 distinct triple hypergeometric functions whose names are X i (i =1,..., 20) to investigate their twenty Laplace inte- gral representations which include the confluent hypergeometric func- tions 0 F 1 , 1 F 1 , a Humbert function Ψ 2 , a Humbert function Φ 2 in their kernels. The Exton functions X i have been studied a lot until today, for example, see [2, 5, 6, 7, 8, 9, 10]. Here, we choose to investigate the Exton function X 5 to present (presumably new) 25 integral representa- tions of Euler type which contain the Exton function X 5 itself, the Exton function X 6 , the Horn’s functions H 3 and H 4 , and the hypergeometric function F = 2 F 1 in their kernels. Received July 12, 2010. Accepted August 10, 2010. 2000 Mathematics Subject Classification. Primary 33C20, 33C65; Secondary 33C05, 33C60, 33C70, 68Q40, 11Y35. Key words and phrases. Generalized hypergeometric series; Multiple hyperge- ometric functions; Integrals of Euler type; Laplace integral; Exton functions Xi ; Humbert function Ψ2; Appell function F4; Srivastava function F (3) .