www.ccsenet.org/jmr Journal of Mathematics Research Vol. 2, No. 3; August 2010 On Certain Hypergeometric Summation Theorems Motivated by the Works of Ramanujan, Chudnovsky and Borwein M. I. Qureshi Department of Applied Sciences and Humanities, Faculty of Engineering and Technology Jamia Millia Islamia (A central university), Jamia Nagar, New Delhi-110025, INDIA E-mail: miqureshi delhi@yahoo.co.in Izharul H. Khan Department of Applied Sciences and Humanities, Faculty of Engineering and Technology Jamia Millia Islamia (A central university), Jamia Nagar, New Delhi-110025, INDIA E-mail: izhargkp@rediffmail.com M. P. Chaudhary (Corresponding author) Department of Applied Sciences and Humanities, Faculty of Engineering and Technology Jamia Millia Islamia (A central university), Jamia Nagar, New Delhi-110025, INDIA E-mail: mpchaudhary 2000@yahoo.com Abstract In the present paper, we obtain numerical values for Gaussian hypergeometric summation theorems by giving particular values to the parameters a, b and the argument x; three summation theorems for 2 F 3 ( 1 4 , 3 4 ; 1 2 , 1 2 , 1; x), three summation the- orems for 4 F 3 ( 1 2 , 1 2 , 1 2 , a+b b ;1, 1, a b ; x), two summation theorems for 4 F 3 ( 1 2 , 1 3 , 2 3 , a+b b ;1, 1, a b ; x), four summation theorems for 4 F 3 ( 1 2 , 1 6 , 5 6 , a+b b ;1, 1, a b ; x) and ten summation theorems for 4 F 3 ( 1 2 , 1 4 , 3 4 , a+b b ;1, 1, a b ; x). A.M.S.(M.O.S.) Subject Classification (1991): 33-Special Functions. Keywords: Ramanujan integrals, Ramanujan series, Ordinary Bessel function of first kind of order n, Borwein and Chudnovsky series 1. Ordinary Bessel Function of First Kind of Order n J n ( x) = ( x/2) n Γ(n + 1) 0 F 1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ; n + 1; − x 2 4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (1) 1.1 Lemma If a, p and n are suitably adjusted real or complex numbers such that associated Pochhammer’s symbols are well- defined, then we have (a + pn) = a ( a+p p ) n ( a p ) n (2) B. C. Berndt ( pp.352-354) listed all of Ramanujan’s series for 1 π found in the 133 unorganized pages of the second and third notebooks of Ramanujan (1984). 1.2 Seventeen Ramanujan’s Series [Berndt, B. C(Part-IV); Hardy, G. H.; Ramanujan, S., 1914; Venkatachala, B. J., 2000] R 4 ≡ 4 π = 1 + 7 4  1 2 3 + 13 4 2  1.3 2.4 3 + 19 4 3  1.3.5 2.4.6 3 + ··· = ∞  n=0 (6n + 1) ( 1 2 ) 3 n 4 n (n!) 3 (3) 196 ISSN 1916-9795 E-ISSN 1916-9809