On Certain Hypergeometric Summation Theorems - CiteSeerX

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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 [email protected] 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 [email protected] 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 F3 ( 14 , 34 ; 12 , 12 , 1; x), three summation thea 1 1 2 a+b a orems for 4 F3 ( 12 , 12 , 12 , a+b b ; 1, 1, b ; x), two summation theorems for 4 F 3 ( 2 , 3 , 3 , b ; 1, 1, b ; x), four summation theorems 1 1 5 a+b a 1 1 3 a+b a for 4 F3 ( 2 , 6 , 6 , b ; 1, 1, b ; x) and ten summation theorems for 4 F3 ( 2 , 4 , 4 , b ; 1, 1, 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 ⎤ ⎡ ⎥ ⎢⎢⎢ ; x2 ⎥⎥⎥⎥ (x/2)n ⎢⎢⎢ Jn (x) = − ⎥⎥⎥ 0 F1 ⎢ ⎢⎣ Γ(n + 1) 4⎦ n+1;

(1)

1.1 Lemma If a, p and n are suitably adjusted real or complex numbers such that associated Pochhammer’s symbols are welldefined, then we have C a+p D

(a + pn) =

a

p

n

CaD

(2)

p n

B. C. Berndt ( pp.352-354) listed all of Ramanujan’s series for third notebooks of Ramanujan (1984).

1 π

found in the 133 unorganized pages of the second and

1.2 Seventeen Ramanujan’s Series [Berndt, B. C(Part-IV); Hardy, G. H.; Ramanujan, S., 1914; Venkatachala, B. J., 2000] R4 ≡ =

4 7 1 = 1+ π 4 2

+

13 1.3 42 2.4

3

+

19 1.3.5 43 2.4.6

3

+ ···

3 ∞ (6n + 1) ( 1 ) 2 n

n=0

196

3

(3)

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R5 ≡

Journal of Mathematics Research 16 47 1 = 5+ π 64 2 =

3

+

89 1.3 642 2.4

3

+

131 1.3.5 643 2.4.6

3


+ ···

3 ∞ (42n + 5) ( 1 ) 2 n 3

n

n=0

(4)

64 (n!)

47 √5 + 29 1 3 √5 − 1 8 √ 32 R6 ≡ + = (5 5 − 1) + π 64 2 2 89 √5 + 59 1.3 3 √5 − 1 16 + + ··· 2.4 2 642 √ √ 3 √ 8n ∞ (42 5n + 5 5 + 30n − 1) ( 12 )n 5 − 1 = 2 64n (n!)3 n=0 R7 ≡

1.3 1.4 2.5 2 27 1 1 22 + 32 = 2 + 17 4π 2 3 3 27 2.4 3.6 3.6 27 1 1 ∞ (15n + 2) ( 2 )n ( 3 )n ( 23 )n 2 n = 27 (n!)3 n=0

2

(5)

+ ··· (6)

√ 1.3 1.4 2.5 4 15 3 1 1 2 4 + 70 R8 ≡ = 4 + 37 2π 2 3 3 125 2.4 3.6 3.6 125 ∞ (33n + 4) ( 12 )n ( 13 )n ( 23 )n 4 n = 125 (n!)3 n=0

2

√ 1.3 1.7 5.11 4 1 1 5 4 5 5 + 23 R9 ≡ √ = 1 + 12 2 6 6 125 2.4 6.12 6.12 125 2π 3 ∞ (11n + 1) ( 12 )n ( 16 )n ( 56 )n 4 n = 125 (n!)3 n=0

R10 ≡

√ 1 1 54 85 85 = 8 + 141 √ 2 6 6 85 18π 3 =

R11 ≡

3

+ 274

2

+ ···

∞ (133n + 8) ( 12 )n ( 16 )n ( 56 )n 4 85 (n!)3 n=0

(8)

6

+ ···

n

(9)

(10)

3 4 59 1.3 1.3.5.7 31 1 1.3 + 2 5 − ··· √ = − 3 2 42 4 3.4 3 .4 2.4 42 .82 π 3 =

∞ (−1)n (28n + 3) ( 12 )n ( 14 )n ( 34 )n n=0

R13 ≡

(7)

1.3 1.7 5.11 4 2.4 6.12 6.12 85

4 3 23 1 1.3 43 1.3 1.3.5.7 + 5 − ··· = − 3 π 2 2 2 42 2 2.4 42 .82 ∞ (−1)n (20n + 3) ( 21 )n ( 14 )n ( 34 )n = (n!)3 22n+1 n=0

R12 ≡

+ ···

(n!)3 3n 4n+1

(11)

4 23 283 1 1.3 543 1.3 1.3.5.7 + 5 − ··· = − π 18 183 2 42 18 2.4 42 .82

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197

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Journal of Mathematics Research ∞ (−1)n (260n + 23) ( 12 )n ( 14 )n ( 34 )n

=

n=0

R14 ≡

∞ (−1)n (644n + 41) ( 12 )n ( 14 )n ( 34 )n n=0

R16 ≡

R17 ≡

(15)

1 1 11 1 1.3 21 1.3 1.3.5.7 + 5 + ··· √ = + 3 9 9 2 42 9 2.4 42 .82 2π 2 ∞ (10n + 1) ( 12 )n ( 14 )n ( 43 )n

(16)

(n!)3 92n+1

3 1 83 1.3 1.3.5.7 43 1 1.3 + 5 + ··· + √ = 49 493 2 42 49 2.4 42 .82 3π 3 =

∞ (40n + 3) ( 12 )n ( 14 )n ( 34 )n n=0

(17)

(n!)3 (49)2n+1

2 19 299 1 1.3 579 1.3 1.3.5.7 + 5 + ··· = + √ 99 993 2 42 99 2.4 42 .82 π 11 =

∞ (280n + 19) ( 12 )n ( 14 )n ( 34 )n n=0

R20 ≡

(14)

√ 2 3 9 1 1.3 17 1.3 1.3.5.7 + 2 + ··· = 1+ π 9 2 42 9 2.4 42 .82 ∞ (8n + 1) ( 12 )n ( 14 )n ( 34 )n = (n!)3 9n n=0

n=0

R19 ≡

(13)

(n!)3 5n (72)2n+1

4 1123 22583 1 1.3 44043 1.3 1.3.5.7 + − ··· = − π 882 8823 2 42 8825 2.4 42 .82 ∞ (−1)n (21460n + 1123) ( 12 )n ( 14 )n ( 34 )n = (n!)3 (882)2n+1 n=0

=

R18 ≡

(12)

(n!)3 (18)2n+1

41 4 1329 1.3 1.3.5.7 685 1 1.3 + − ··· − √ = 3 2 42 72 5.72 52 .725 2.4 42 .82 π 5 =

R15 ≡


(18)

(n!)3 (99)2n+1

1 1103 27493 1 1.3 53883 1.3 1.3.5.7 + + + ··· √ = 992 9910 2.4 42 .82 996 2 42 2π 2 =

∞ (26390n + 1103) ( 12 )n ( 14 )n ( 34 )n n=0

(19)

(n!)3 (99)4n+2

The first three series representations R4 , R5 and R6 for π1 are found in Ramanujan’s papers [Hardy, G. H., 2000, pp.36-37; Ramanujan, S., 1914]. The remaining fourteen series representations R7 , R8 , · · · , R20 are stated without proof in Ramanujan’s papers [Hardy, G. H., 2000, pp.37-38; Ramanujan, S., 1914]. 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. 1.3 Borwein Series [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 R20 to calculate π to 17 million places and found that the formula converges on 198

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the exact value with far greater efficiency than any previous method. This success proved that Ramanujan’s insight was correct. (−1)n (6n)! 1 × = 12 3 π n=0 (n!) (3n)! ∞

R21 ≡ ×

√ √ [212175710912 61 + 1657145277365 + n(13773980892672 61 + 107578229802750)] √ (3n+ 3 ) {5280(236674 + 30303 61)} 2

(20)

1.4 Chudnovsky Series (Chudnovsky, D. V., 1988) In 1989, David and Gregory Chudnovsky computed 1011196961 digits of π, using the series [Berndt, B. C., 2003, p.201] (−1)n (6n)! (13591409 + n545140134) 1 = 12 (n+ 1 ) π n=0 (n!)3 (3n)! {(640320)3 } 2 ∞

R22 ≡

(21)

which is similar to those found by Ramanujan. 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 R4 to R20 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, B. C (Part-II), p.8]. All formulas are stated by writing out the first few terms in each series. 2. Main Summation Theorems

⎡ ⎢⎢⎢ ⎢⎢ F ⎢⎢⎣ 2 3⎢ ⎡ ⎢⎢⎢ ⎢⎢ F ⎢⎢⎣ 2 3⎢

1 1 2, 2, 1 3 4, 4

1 1 2, 2,

⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 2 F3 ⎢

1 1 2, 2,

⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 4 F3 ⎢

⎤ ⎥ 1 π −π2 ⎥⎥⎥⎥ ⎥⎥⎥ = J0 9 ⎦ 2 3

;

1 1 1 7 2, 2, 2, 6

1, 1,

1, 1, ; ;

5 42


;

; ;

⎤ ⎥ 1 ⎥⎥⎥⎥ 4 ⎥⎥ = 4 ⎥⎦ π ⎤ ⎥ 1 ⎥⎥⎥⎥ 16 ⎥⎥ = 64 ⎥⎦ 5π

⎤ √ √ ⎥⎥ (8 + 40 5) (47 − 21 5) ⎥⎥⎥⎥ ⎥⎥⎥ = ⎥⎦ 128 31π

1 1 2 17 2 , 3 , 3 , 15

1, 1,

;

1 6

1 1 1 47 2 , 2 , 2 , 42

√ (45−8 5) 330

⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 4 F3 ⎢

⎤ ⎥⎥⎥ ⎥ − π2 ⎥⎥⎥⎥ = − J0 (π) ⎦

;

1;

√ 1 1 1 (375−8 5) , , , 2 2 2 330

1, 1,

⎤ ⎥ −π2 ⎥⎥⎥⎥ ⎥⎥ = 0 4 ⎥⎦

;

1;

1;

1 3 4, 4

⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 4 F3 ⎢

⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ 4 F3 ⎢ ⎣

1 3 4, 4

2 15

; ;

⎤ ⎥ 2 ⎥⎥⎥⎥ 27 ⎥⎥ = 27 ⎥⎦ 8π

(22)

(23)

(24)

(25)

(26)

(27)

(28)

199

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The equation (28) is known hypergeometric form of Berndt and Rankin [Berndt, B. C, 2003, p.298(3.4)]. ⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 4 F3 ⎢

1 1 2 37 2 , 3 , 3 , 33

⎡ ⎢⎢⎢ ⎢⎢ F ⎢⎢⎣ 4 3⎢ ⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 4 F3 ⎢

1 1 5 141 2 , 6 , 6 , 133

1 1 3 23 2 , 4 , 4 , 20

1 1 3 283 2 , 4 , 4 , 260

;

;

1 1 3 22583 2 , 4 , 4 , 21460

;

⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 4 F3 ⎢

⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 4 F3 ⎢ 200

1123 21460

1 1 3 11 2 , 4 , 4 , 10 1 10

1 1 3 299 2 , 4 , 4 , 280

1, 1,

19 280

1 1 3 27493 2 , 4 , 4 , 26390

1, 1,

1103 26390

;

3 40

; ;

;

; ;

⎤ ⎥ 8 −1 ⎥⎥⎥⎥ ⎥⎥ = 4 ⎥⎦ 3π

(32)

(33)

⎤ ⎥ 72 −1 ⎥⎥⎥⎥ ⎥⎥ = 324 ⎥⎦ 23π

;

⎤ √ ⎥ 1 ⎥⎥⎥⎥ 2 3 ⎥⎥⎥ = 9⎦ π

;

⎤ √ ⎥ 1 ⎥⎥⎥⎥ 9 2 ⎥⎥ = 81 ⎥⎦ 4π

;

1 8

1 1 3 43 2 , 4 , 4 , 40

(31)

(34)

(35)

⎤ ⎥ 3528 −1 ⎥⎥⎥⎥ ⎥⎥ = 777924 ⎥⎦ 1123π

;

1, 1,

1, 1,

(30)

⎤ √ ⎥ −1 ⎥⎥⎥⎥ 288 5 ⎥⎥ = 25920 ⎥⎦ 205π

1 1 3 9 2, 4, 4, 8

1, 1,

;

;

41 644

1, 1,

1, 1,

⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 4 F3 ⎢

23 260

1, 1,

(29)

⎤ √ ⎥ 16 3 −1 ⎥⎥⎥⎥ ⎥⎥ = 12 ⎥⎦ 9π

; ;

1 1 3 685 2 , 4 , 4 , 644

⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 4 F3 ⎢

;

3 28

1, 1,

⎡ ⎢⎢⎢ ⎢⎢ F ⎢⎢⎣ 4 3⎢

;

3 20

1, 1,

1 1 3 31 2 , 4 , 4 , 28

⎡ ⎢⎢⎢ ⎢⎢ F ⎢⎢⎣ 4 3⎢

;

⎤ √ ⎥ 4 ⎥⎥⎥⎥ 85 255 ⎥⎥ = 85 ⎥⎦ 432π

;

8 133

1, 1,

;

⎤ √ ⎥ 4 ⎥⎥⎥⎥ 5 15 ⎥⎥⎥ = 125 ⎦ 6π

;

1 11

1, 1,

⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 4 F3 ⎢

⎡ ⎢⎢⎢ ⎢⎢ F ⎢⎢⎣ 4 3⎢

;

1 1 5 12 2 , 6 , 6 , 11

⎡ ⎢⎢⎢ ⎢⎢ F ⎢⎢⎣ 4 3⎢

⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 4 F3 ⎢

4 33

1, 1,

⎤ √ ⎥ 15 3 4 ⎥⎥⎥⎥ ⎥⎥ = 125 ⎥⎦ 8π

;

;

(36)

(37)

(38)

⎤ √ ⎥ 49 3 1 ⎥⎥⎥⎥ ⎥⎥⎥ = 2401 ⎦ 27π

(39)

⎤ √ ⎥ 1 ⎥⎥⎥⎥ 198 11 ⎥⎥ = 9801 ⎥⎦ 209π

(40)

⎤ √ ⎥⎥⎥ 9801 2 1 ⎥⎥⎥ ⎥ = 96059601 ⎥⎦ 4412π

(41)

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The equation (41) is the correct form of erroneous hypergeometric form of Berndt and Rankin[2003, p.298(3.5)]. The series (28) and (41) are given in sections 13 and 14 of (Ramanujan, S., 1914) and [Ramanujan, S., 1984, p.378]. ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ 4 F3 ⎢ ⎢⎣

1 6

,

1 2

,

5 6

√ 13986156603584 √61+109235375080115 13773980892672 61+107578229802750

,

;

−1728

√ √ 212175710912 √61+1657145277365 {5280(236674 + 30303 61)} 1 , 1 , 13773980892672 ; 61+107578229802750 3 √ {5280(236674 + 30303 61)} 2 = √ 12 (212175710912 61 + 1657145277365) π

⎡ ⎢⎢⎢ ⎢⎢ ⎢⎢⎣ 4 F3 ⎢

1 6

,

1 2

,

5 6

,

558731543 545140134

;

13591409 1 , 1 , 545140134

⎤ 3 ⎥ −1728 ⎥⎥⎥⎥ (640320) 2 ⎥⎥ = 3⎥ ⎦ 12 (13591409) π ; (640320)

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥ 3⎥ ⎥⎥⎦

(42)

(43)

Above summation theorems for 4 F3 are not available in the mostly available literature of special functions including the monumental works of Prudnikov et. al. [Prudnikov, A. P., 1990, pp.552-563]. The summation theorems presented by equations (42) and (43) are the hypergeometric forms of the Borwein series (20) and Chudnovsky series (21) respectively. 3. Proof of (22) to (43) The hypergeometric summation theorems from (22) to (43) can be established from the results R1 to R22 respectively. Proof of (24) : To prove (24), we shall consider the Ramanujan’s integral R3 .

π 2

0

Using Maclaurin’s series for cos x =

π 2

0

π 2π sin2 θ π sin θ 1 2 dθ = dθ cos cos 3 2 0 3

∞ n=0

(−1)n x2n (2n)! ,

we get

2 ∞ θ (−1)n ( 2π sin ) 3 (2n)! n=0

2n

1 dθ = 2

0

π 2

2n ∞ (−1)n ( π sin θ ) 3

(2n)!

n=0

dθ

Now interchanging the order of summation and integration, we get 2 n π ∞ ∞ n π2 n π 2 2 (−1)n ( 4π9 ) 1 (−1) ( 9 ) 4n sin θ dθ = sin2n θ dθ (2n)! 2 (2n)! 0 0 n=0 n=0

n

2 ∞ 1 (−1)n ( 4π9 ) Γ( 4n+1 2 ) Γ( 2 )

(2n)! 2 Γ( 4n+2 2 )

n=0

On simplifying, we get

n

∞ n π 2n+1 1 1 (−1) ( 9 ) Γ( 2 ) Γ( 2 ) = 2 n=0 (2n)! 2 Γ( 2n+2 2 ) 2

n

∞ π2 n 1 (− 36 ) 1 1 2 n=0 (1)n n! n=0 ( 2 )n ( 2 )n (1)n n! ⎤ ⎤ ⎡ ⎡ 1 3 ⎢⎢⎢ ⎢⎢⎢ ; , 4; 2⎥ 2⎥ ⎥ ⎥⎥⎥ 4 ⎥ 1 −π −π ⎥⎥⎥ ⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎦ = ⎢⎢⎣ 2 F3 ⎢ 0 F1 ⎢ ⎥ ⎥⎦ ⎢ ⎣ 9 2 36 1 1 1; 2, 2, 1 ; 2 ∞ ( 14 )n ( 34 )n (− π9 )

=

(44)

Now write the right hand side of (44) in terms of ordinary Bessel function of order zero, we get the summation theorem (24). Similarly, we can prove (22) and (23). Now we shall establish (27) from Ramanujan’s series R6 . √ √ 3 √ ∞ (42 5n + 5 5 + 30n − 1) ( 12 )n 5 − 1 n=0

64n (n!)3

2

8n

=

32 π

√ √ √ 8 n ∞ {(−1 + 5 5) + (30 + 42 5)n} ( 12 )n ( 12 )n ( 12 )n = ( 5 − 1) > n=0

(1)n (1)n n!


214

=

32 π

(45)

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Now using the Lemma (2) in (45), we get √ √ √ 8 >n ∞ (−1 + 5 5) ( 29+47 √5 ) ( 1 ) ( 1 ) ( 1 ) = 5) 32 30+42 5 n 2 n 2 n 2 n (−1 + = √ 14 −1+5 √5 π 2 (1) (1) ( ) n! n=0 n

⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ 4 F3 ⎢ ⎢⎣

or

n

1 1 1 2, 2, 2,

⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ 4 F3 ⎢ ⎣

30+42 5 n √ 29+47 √5 ; 30+42 5 (−1

1, 1,

√ −1+5 √5 30+42 5

1 1 1 2, 2, 2,

;

√ (375−8 5) 330

1, 1,

√ (45−8 5) 330

; ;

⎤ √ 8 ⎥⎥⎥ + 5) ⎥⎥⎥⎥ 32 ⎥⎥⎥ = √ 14 2 (−1 + 5 5)π ⎦⎥

⎤ √ ⎥⎥ √ (47 − 21 5) ⎥⎥⎥⎥ (8 + 40 5) ⎥⎥⎥ = ⎥⎦ 128 31π

(46)

which is the summation theorem (27). Similarly, in view of the Lemma (2), we can establish the remaining summation theorems (25)-(43), from Ramanujan series R4 , R5 , R7 , R8 , R9 , · · · , R20 , Borwein series R21 and Chudnovsky series R22 respectively. 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 R4 to R20 . For example, in 1987, R. Wm. Gosper employed the series R20 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 H. Chan (Berndt, B. C, 2001), who derived a series yielding 73 or 74 digits of π per term.

1 π

is held by Berndt and H.

Acknowledgement The authors are thankful to Prof. R. Y. Denis and Prof. M. A. Pathan for their valuable suggestions during the preparation of the manuscript of this research paper. Second and third authors are also grateful to the referees for their comments for possible improvements in the text of the manuscript, both the authors are highly thankful to the Editor-JMR, for his kindness and coperations for finally publication of this paper. References Berndt, B. C. (1985, 1989, 1991, 1994, 1998). Ramanujan’s notebooks. Parts.I-V, Springer-Verlag, New York. Berndt, B. C. and Chan, H. H. (2001). Eisenstein series and approximations to π. Illinois J. Math, 45, 75-90. Berndt, B. C. and Rankin, R. A. (2003). Ramanujan: Essays and Surveys. Hindustan Book Agency (India), New Delhi, Indian Edition. Borwein, J. M. and Borwein, P. B. (1987). Pi and the AGM. Can. Math. Soc., John Wiley, New york. Borwein, J. M. and Borwein, P. B. (1987). Ramanujan’s rational and algebraic series for π1 . Indian J. Math, 51, 147-160. Borwein, J. M. and Borwein, P. B. (1988). More Ramanujan-type series for π1 , in Ramanujan Revisited. Academic Press, Boston, 359-374. Borwein, J. M. and Borwein, P. B. (1993). Class number three Ramanujan-type series for π1 . J. Comput. Appl. Math, 46, 281-290. Chudnovsky, D. V. and Chudnovsky, G. V. (1988). Approximations and complex multiplication according to Ramanujan, in Ramanujan Revisited. Academic Press, Boston, 375-472. Hardy, G. H., Aiyar, P. V. Seshu and Wilson, B. M. (1962). Collected papers of Srinivasa Ramanujan. First published by Cambridge University Press, Cambridge, 1927; Reprinted by Chelsea, New York; Reprinted by the American Mathematical Society, Providence, RI, 2000. Preston, R. (1992). Profiles: The mountains of π. The New Yorker (March 2,), 36-67. Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I. (1986). Integrals and series Vol. 3: More special functions. Nauka, Moscow. Translated from the Russian by G. G. Gould, Gordon and Breach Science Publishers, New York, Philadelphia, London, Paris, Montreux, Tokyo, Melbourne, 1990. Ramanujan, S. (1914). Modular equations and approximations to π. Quart. J. Math, 45, 350-372. Ramanujan, S. (1984). Notebooks of Srinivasa Ramanujan. Vol. II, Tata Institute of Fundamental Research, Bombay, 1957; Reprinted by Narosa, New Delhi. Venkatachala, B. J., Vinay, V. and Yogananda, C. S. (2000). Ramanujan’s papers. Prism Books Pvt. Ltd., Bangalore, Mumbai. 202

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