Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications 60 Some cases of reducible Generalized Hypergeometric Functions Awadhesh Kumar Pandey,Ramakant Bhardwaj ∗ ,Kamal Wadhwa ∗∗ and Nitesh Singh Thakur ∗∗∗ Department of Mathematics,Patel Institute of Technology, Bhopal, M.P. *Truba Institute of Technology, Bhopal, M.P. **Govt. Narmda P.G. College, Hoshangabad, M.P. ***Patel College of Science & Technology, Bhopal, M.P. Abstract In this paper we consider the integrals of Generalized hypergeometric function of three variables given by Saran Shanti [4] and obtained functions of two variables of Horn’s list given in ErdelyiA.[1]. Our results are also motivated by Singh Pooja & Singh Prof. (Dr.) Harish [3]. Corresponding author E.mail:- pandey1172@gmail.com , pandey.awadhesh1972@gmail.com Introduction Horn investigated in particular hypergeometric series of order two and found that , apart from certain series which are either expressible in terms of product of two hypergeometric series in one variable, there are 34 distinct convergent hypergeometric series Erdelyi A [1]. Saran Shanti [4], gave some integral associated with hypergeometric function of three variables. We are using some of them for our investigation such as F E and F K . Recently the method adopting here has been used by Singh Pooja & Singh Prof. (Dr.) Harish [3]. We are giving here same treatment with some modifications. Saran Shanti [4], gave the following summations F E (  ,  ,  ,  ,  ,   ;   ,  ,  ; x, y, z) = ∑ (" # ) %&’&( () # ) % () * ) ’&( +!-!.!(/ # ) % (/ * ) ’ (/ 0 ) ( 1 +,-,.34 5 + 6 - 7 . .… (1.1) F K (  ,  ,  ,  ,  ,  ;  ,  ,  ; x,y, z) = ∑ (" # ) % (" * ) ’&( () # ) %&( () * ) ’ +!-!.!(/ # ) % (/ * ) ’ (/ 0 ) ( 1 +,-,.34 5 + 6 - 7 . .... (1.2) and their integral results are also given by Saran Shanti [4], as F E (  ,  ,  ,  ,  ,   ;   ,  ,  ; x, y, z) = 8/ * 8/ 0 8(9/ * 9/ 0 ) (:;) * ∫ / (−>) 9/ * ( > − 1) 9/ 0 F 2 (  ,  ,  ,  ,  +  -1; x, @ A + C 9A )dt ……. (1.3) where |5| + E @ A + C 9A E < 1 along the contour. F K (  ,  ,  ,  ,  ,   ;   ,  ,  ; x, y, z) = 8F8F # 8(9F9F # ) (:;) * ∫ / (−>) 9F ( > − 1) 9F # 2 F 1 (r,   ;  +   ; G A ) F 2 (  ,   ,H  ,  ,  ;6, C 9A ) dt ….. (1.4) Where b 1 = r + H  – 1, |>| > |5|, E C 9A E < 1 − |6| along the integral. Singh Pooja & Singh Prof. (Dr.) Harish[3], used argument as hyperbolic function in (1.3) & (1.4) as F E (  ,  ,  ,  ,  ,   ;   ,  ,  ; coshx, coshy, coshz) = 8/ * 8/ 0 8(9/ * 9/ 0 ) (:;) * ∫ / (−>) 9/ * ( > − 1) 9/ 0 F 2 (  ,  ,  ,  ,  +  -1; coshx, /KLM@ A + /KLMC 9A )dt ……. (1.5) F K (  ,  ,  ,  ,  ,   ;   ,  ,  ; coshx, NOℎ6, z) = 8F8F # 8(9F9F # ) (:;) * ∫ / (−>) 9F ( > − 1) 9F # × 2 F 1 (r,  ,   +   ; NOℎ6, RSTUV A ) F 2 (  ;  ,H  ;  ,  ;NOℎ6, /KLMC 9A ) dt ….. (1.6) The following results are due to Singh Pooja & Singh Prof. (Dr.) Harish[3], F E (  ,   ,   ,   ,  +  −1 ,   +  −1 ;   ,  ,  ; coshx, coshy, coshz) = (1 − NOℎ5– NOℎ6 ) 9" # H 1 (1-  ,  ,   +  −1 ,   ; /KLM@ /KLMGX/KLM@9 , /KLMC /KLMGX/KLM@9 ) ...(1.7) F E (  ,  ,  ,  ,  +  −1,  +  −1;   ,  ,  ; coshx, coshy, coshz) =(1 − NOℎ5 – NOℎ6 − NOℎ7 ) 9" #