On Certain Hypergeometric Summation Theorems Motivated by the Works of Ramanujan , Chudnovsky and Borwein

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 2F3( 4 , 3 4 ; 1 2 , 1 2 , 1; x), three summation theorems for 4F3( 2 , 1 2 , 1 2 , a+b b ; 1, 1, a b ; x), two summation theorems for 4F3( 1 2 , 1 3 , 2 3 , a+b b ; 1, 1, a b ; x), four summation theorems for 4F3( 2 , 1 6 , 5 6 , a+b b ; 1, 1, a b ; x) and ten summation theorems for 4F3( 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.

1. Ordinary Bessel Function of First Kind of Order n

Lemma
If a, p and n are suitably adjusted real or complex numbers such that associated Pochhammer's symbols are welldefined, then we have (2) B. C. ) listed all of Ramanujan's series for 1 π found in the 133 unorganized pages of the second and third notebooks of Ramanujan (1984).[Berndt, B. C(Part-IV); Hardy, G. H.;Ramanujan, S., 1914;Venkatachala, B. J., 2000
Ramanujan has used the equality sign for his approximations to π but he states explicitly that the formulae are correct only to a certain number of decimal places.The symbol ≈ had not come into vogue by then.[Borwein, J. M., 1987;1988;1993] In 1987, Borwein and Borwein [Borwein, J. M., 1987;Berndt, B. C., 2003, p.193], two computer scientists used a following version of Ramanujan's formula R 20 to calculate π to 17 million places and found that the formula converges on the exact value with far greater efficiency than any previous method.This success proved that Ramanujan's insight was correct.
In 1991, the Chudnovskys computed in excess of 2.16 billion digits.Many details of their work on a largely home-built computer are given in the delightful profile "The Mountains of Pi" (Preston, R., 1992).
The Ramanujan series from R 4 to R 20 were not proved until 1987, when J. M. and P. B. Borwein proved them in their book [Borwein, J. M, 1987, pp.177-187].In three further papers (Borwein, J. M, 1987), (Borwein, J. M, 1988) and (Borwein, J. M, 1993), they established several additional formulas of this type.D. V. and G. V. Chudnovsky (1988) not only also proved formulas of this sort, but they, moreover, found representations for other transcendental constants, some involving gamma functions, by hypergeometric series of the same kin.
It should be remarked that Ramanujan has no notation for hypergeometric series [Berndt,p.8].All formulas are stated by writing out the first few terms in each series.
It may be remarked that the above summations will lead to the excellent value of π approximated to various places of decimals, by taking few terms of the Ramanujan series R 4 to R 20 .For example, in 1987, R. Wm.Gosper employed the series R 20 in calculating 17000000 digits of π.
Currently, the world record for the most digits of π per term in a series of Ramanujan-type for 1 π is held by Berndt and H. H. Chan (Berndt, B. C, 2001), who derived a series yielding 73 or 74 digits of π per term. ]