Extensions of Certain Classical Summation Theorems for the Series

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Volume 2010, Article ID 309503,26pages doi:10.1155/2010/309503

Research Article

Extensions of Certain Classical Summation Theorems for the Series

2

F

1

,

3

F

2

, and

4

F

3

with Applications in Ramanujan’s Summations

Yong Sup Kim,

¹

Medhat A. Rakha,

^{2, 3}

and Arjun K. Rathie

⁴

1Department of Mathematics Education, Wonkwang University, Iksan 570-749, Republic of Korea

2Mathematics Department, College of Science, Suez Canal University, Ismailia 41522, Egypt

3Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Muscat, Alkhod 123, Oman

4Vedant College of Engineering and Technology, Village-Tulsi, Post-Jakhmund, Bundi, Rajasthan State 323021, India

Correspondence should be addressed to Medhat A. Rakha,medhat@squ.edu.om Received 20 May 2010; Revised 7 September 2010; Accepted 23 September 2010 Academic Editor: Teodor Bulboac˘a

Copyrightq2010 Yong Sup Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Motivated by the extension of classical Gauss’s summation theorem for the series2F1given in the literature, the authors aim at presenting the extensions of various other classical summation theorems such as those of Kummer, Gauss’s second, and Bailey for the series2F1, Watson, Dixon and Whipple for the series3F2, and a few other hypergeometric identities for the series3F2and4F3. As applications, certain very interesting summations due to Ramanujan have been generalized.

The results derived in this paper are simple, interesting, easily established, and may be useful.

1. Introduction

In 1812, Gauss1systematically discussed the series ∞

n0

a_nb_n c_n

zⁿ

n! 1a·b

1·czaa1bb1

1·2cc1 z²· · ·, 1.1 whereλ_ndenotes the Pochhammer symbol definedforλ∈Cby

λ_n:

⎧⎨

⎩

1 n0

λλ1· · ·λn−1 n∈N:{1,2,3, . . .}. 1.2

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It is noted that the series 1.1 and its natural generalization _PFq in 1.6 are of great importance to mathematicians and physicists. This series1.1has been known as the Gauss series or the ordinary hypergeometric series and may be regarded as a generalization of the elementary geometric series. In fact1.1reduces to the geometric series in two cases, when a candb 1 also whenb canda 1. The series1.1is represented by the notation

2F1a, b;c;zor

2F1

⎡

⎢⎢

⎣ a, b

; z c

⎤

⎥⎥

⎦, 1.3

which is usually referred to as Gauss hypergeometric function. In1.1, the three elementsa, b, andcare described as the parameters of the series, andzis called the variable of the series.

All four of these quantities may be real or complex with an exception thatcis neither zero nor a negative integer. Also, in1.1, it is easy to see that if any one of the numerator parameters aorbor both is a negative integer, then the series reduces to a polynomials, that is, the series terminates.

The series1.1is absolutely convergent within the unit circle when|z|< 1 provided thatc /0,−1,−2, . . .. Also when|z|1, the series is absolutely convergent ifRc−a−b>0, conditionally convergent if−1<Rc−a−b≤0,z /1 and divergent ifRc−a−b≤ −1.

Further, if in1.1, we replacezbyz/band letb → ∞, thenb_nzⁿ/bⁿ → zⁿ, and we arrive to the following Kummer’s series

∞ n0

a_n c_n

zⁿ

n! 1 a

1·cz aa1

1·2cc1z²· · ·. 1.4 This series is absolutely convergent for all values of a, c, and z, real or complex, excludingc0,−1,−2, . . .and is represented by the notation1F1a;c;zor

1F1

⎡

⎢⎢

⎣ a,

; z c

⎤

⎥⎥

⎦, 1.5

which is called a confluent hypergeometric function.

Gauss hypergeometric function 2F1 and its confluent case 1F1 form the core special functions and include, as their special cases, most of the commonly used functions. Thus₂F1

includes, as its special cases, Legendre function, the incomplete beta function, the complete elliptic functions of first and second kinds, and most of the classical orthogonal polynomials.

On the other hand, the confluent hypergeometric function includes, as its special cases, Bessel functions, parabolic cylindrical functions, and Coulomb wave function.

Also, the Whittaker functions are slightly modified forms of confluent hypergeometric functions. On account of their usefulness, the functions ₂F1 and ₁F1 have already been explored to considerable extent by a number of eminent mathematicians, for example, C. F.

Gauss, E. E. Kummer, S. Pincherle, H. Mellin, E. W. Barnes, L. J. Slater, Y. L. Luke, A. Erd´elyi, and H. Exton.

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A natural generalization of2F1 is the generalized hypergeometric seriespFq defined by

pFq

⎡

⎢⎢

⎣

a1 · · · ap

; z b1 · · · bq

⎤

⎥⎥

⎦^∞

n0

a1_n· · · ap

n

b1_n· · · bq

n

zⁿ

n!. 1.6

The series1.6is convergent for all|z|<∞ifp≤qand for|z|<1 ifpq1 while it is divergent for allz,z /0 ifp > q1. When|z|1 withpq1, the series1.6converges absolutely if

R

⎛

⎝^q

j1

bj− p j1

aj

⎞

⎠>0, 1.7

conditionally convergent if

−1<R

⎛

⎝^q

j1

bj− p j1

aj

⎞

⎠≤0, z /1 1.8

and divergent if

R

⎛

⎝^q

j1

bj− p j1

aj

⎞

⎠≤ −1. 1.9

It should be remarked here that whenever hypergeometric and generalized hypergeometric functions can be summed to be expressed in terms of Gamma functions, the results are very important from a theoretical and an applicable point of view. Only a few summation theorems are available in the literature and it is well known that the classical summation theorems such as of Gauss, Gauss’s second, Kummer, and Bailey for the series

2F1, and Watson, Dixon, and Whipple for the series 3F2 play an important role in the theory of generalized hypergeometric series. It has been pointed out by Berndt2, that very interesting summations due to Ramanujan can be obtained quite simply by employing the above mentioned classical summation theorems. Also, in a well-known paper by Bailey3, a large number of very interesting results involving products of generalized hypergeometric series have been developed. In4a generalization of Kummer’s second theorem was given from which the well-known Preece identity and a well-known quadratic transformation due to Kummer were derived.

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2. Known Classical Summation Theorems

As already mentioned that the classical summation theorems such as those of Gauss, Kummer, Gauss’s second, and Bailey for the series₂F1 and Watson, Dixon, and Whipple for the series3F2 play an important role in the theory of hypergeometric series. These theorems are included in this section so that the paper may be self-contained.

In this section, we will mention classical summation theorems for the series₂F1and

3F2. These are the following.

Gauss theorem5:

2F1

⎡

⎢⎢

⎣ a, b

; 1 c

⎤

⎥⎥

⎦ ΓcΓc−a−b

Γc−aΓc−b 2.1

providedRc−a−b>0.

Kummer theorem5:

2F1

⎡

⎢⎢

⎣

a, b

; −1 1a−b

⎤

⎥⎥

⎦ Γ1a−bΓ1 1/2a

Γ1 1/2a−bΓ1a. 2.2

Gauss’s second theorem5:

2F1

⎡

⎢⎢

⎣

a, b

; 1 1 2

2ab1

⎤

⎥⎥

⎦ Γ1/2Γ1/2a 1/2b 1/2

Γ1/2a 1/2Γ1/2b 1/2. 2.3

Bailey theorem5:

2F1

⎡

⎢⎢

⎢⎣

a, 1−a

; 1 2 c

⎤

⎥⎥

⎥⎦ Γ1/2cΓ1/2c 1/2

Γ1/2c 1/2aΓ1/2c−1/2a 1/2. 2.4

Watson theorem5:

3F2

⎡

⎢⎢

⎢⎣

a, b, c

; 1 1

2ab1, 2c

⎤

⎥⎥

⎥⎦

Γ1/2Γc 1/2Γ1/2a 1/2b 1/2Γc−1/2a−1/2b 1/2

Γ1/2a 1/2Γ1/2b 1/2Γc−1/2a 1/2Γc−1/2b 1/2 2.5

providedR2c−a−b>−1.

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Dixon theorem5:

3F2

⎡

⎢⎢

⎣

a, b, c

; 1 1a−b, 1a−c

⎤

⎥⎥

⎦

Γ1 1/2aΓ1a−bΓ1a−cΓ1 1/2a−b−c Γ1aΓ1 1/2a−bΓ1 1/2a−cΓ1a−b−c

2.6

providedRa−2b−2c>−2.

Whipple theorem5:

3F2

⎡

⎢⎢

⎣

a, b, c

; 1 e, f

⎤

⎥⎥

⎦

πΓeΓ f 2^2c−1Γ1/2a 1/2eΓ

1/2a 1/2f

Γ1/2b 1/2eΓ

1/2b 1/2f 2.7

providedRc>0 andRef−a−b−c>0 withab1 andef2c1.

Other hypergeometric identities5:

3F2

⎡

⎢⎢

⎣

a, 11

2a, b

; 1 1

2a, 1a−b

⎤

⎥⎥

⎦ Γ1a−bΓ1/2a 1/2

Γ1aΓ1/2a−b 1/2, 2.8

4F3

⎡

⎢⎢

⎣

a, 11

2a, b, c

; 1 1

2a, 1a−b, 1a−c

⎤

⎥⎥

⎦

Γ1a−bΓ1a−cΓ1/2a 1/2Γ1/2a−b−c 1/2 Γ1aΓ1a−b−cΓ1/2a−b 1/2Γ1/2a−c 1/2

2.9

It is not out of place to mention here that Ramanujan independently discovered a great number of the primary classical summation theorems in the theory of hypergeometric series. In particular, he rediscovered well-known summation theorems of Gauss, Kummer, Dougall, Dixon, Saalsch ¨utz, and Thomae as well as special cases of the well-known Whipple’s transformation. Unfortunately, Ramanujan left us little knowledge as to know how he made his beautiful discoveries about hypergeometric series.

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3. Ramanujan’s Summations

The classical summation theorems mentioned in Section 2 have wide applications in the theory of generalized hypergeometric series and other connected areas. It has been pointed out by Berndt2that a large number of very interesting summations due to Ramanujan can be obtained quite simply by employing the above mentioned theorems.

We now mention here certain very interesting summations by Ramanujan2.

iForRx>1/2,

1−x−1

x1x−1x−2

x1x2− · · · x

2x−1, 3.1

1− 1

2 ₂

1·3

2·4 ₂

−

1·3·5 2·4·6

₂ · · ·

√π

√2Γ²3/4. 3.2

iiForRx>0,

1x−1

x1x−1x−2

x1x2· · · 2^2x−1Γ²x1

Γ2x1 , 3.3

1 1 2

1 2

₂ 1

2² 1·3

2·4 ₂

1 2³

1·3·5 2·4·6

₂ · · ·

√π

Γ²3/4. 3.4

iiiForRx>0,

1−1 3

x−1 x1 1

5

x−1x−2

x1x2− · · · 2^4xΓ⁴x1

4xΓ²2x1. 3.5

ivForRx>1/4,

1 x−1² x1²

x−1x−2 x1x2

2

· · · 2x 4x−1

Γ⁴x1Γ4x1

Γ⁴2x1 . 3.6

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vForRx>1,

1−3x−1

x15x−1x−2

x1x2− · · ·0, 3.7

11 5

1 2

₂ 1

9 1·3

2·4 ₂

· · · π² 4Γ⁴3/4, 1 1

5² 1

2

1 9²

1·3 2·4

· · · π^5/2 8√

2Γ²3/4, 1

1 2

₂

1·3 2·4

₂

· · · π Γ⁴3/4.

3.8

viForRx<2/3,

1x 1!

3

xx1 2!

₃

· · · 6 sinπx/2sinπxΓ³1/2x1

π²x²Γ3/2x112 cosπx . 3.9

viiForRx>1/2,

13x−1

x15x−1x−2

x1x2· · ·x. 3.10

viiiForRx>1/2,

13

x−1 x1

₂ 5

x−1x−2 x1x2

₂

· · · x²

2x−1. 3.11

We now come to the derivations of these summation in brief.

It is easy to see that the series3.1corresponds to

2F1

⎡

⎢⎢

⎣

1, 1−x

; 1 1x

⎤

⎥⎥

⎦ 3.12

which is a special case of Gauss’s summation theorem2.1fora1,b1−xandc1x.

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The series3.2corresponds to

2F1

⎡

⎢⎢

⎢⎣ 1 2, 1

2

; −1 1

⎤

⎥⎥

⎥⎦ 3.13

which is a special case of Kummer’s summation theorem2.2forab1/2. Similarly the series3.3corresponds to

2F1

⎡

⎢⎢

⎣

1, 1−x

; −1 1x

⎤

⎥⎥

⎦ 3.14

which is a special case of Kummer’s summation theorem2.2fora1,b1−x.

2F1

⎡

⎢⎢

⎣ 1 2, 1

2

; 1 2 1

⎤

⎥⎥

⎦ 3.15

which is a special case of Gauss’s second summation theorem2.3forab1/2 or Bailey’s summation theorem2.4fora1/2 andc1.

Also, it can easily be seen that the series 3.5 to 3.9 correspond to each of the following series:

3F2

⎡

⎢⎢

⎣ 1, 1

2, 1−x

; 1 3

2, 1x

⎤

⎥⎥

⎦, 3F2

⎡

⎢⎣

1, 1−x, 1−x

; 1 1x, 1x

⎤

⎥⎦, 3F2

⎡

⎢⎢

⎣ 1, 3

2, 1−x

; 1 1

2, 1x

⎤

⎥⎥

⎦,

3F2

⎡

⎢⎢

⎣ 1 2, 1

2, 1 4

; 1 1, 5

4

⎤

⎥⎥

⎦, 3F2

⎡

⎢⎢

⎣ 1 2, 1

4, 1 4

; 1 5

4, 5 4

⎤

⎥⎥

⎦,

3F2

⎡

⎢⎢

⎢⎣ 1 2, 1

2, 1 2

; 1 1, 1

⎤

⎥⎥

⎥⎦, 3F2

⎡

⎢⎢

⎣

x, x, x

; 1 1, 1

⎤

⎥⎥

⎦,

3.16 which are special cases of classical Dixon’s theorem2.6foria1,b1/2,c1−x,ii a 1,b c 1−x,iiia 1,b 3/2,c 1−x,iva b 1/2,c 1/4,va 1/2, bc1/4,viabc1/2, andviiabcx, respectively.

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The series3.10which corresponds to

3F2

⎡

⎢⎢

⎣ 1, 3

2, 1−x

; −1 1

2, 1x

⎤

⎥⎥

⎦ 3.17

is a special case of2.8fora1,b1−x, and the series3.11which corresponds to

4F3

⎡

⎢⎢

⎣ 1, 3

2, 1−x, 1−x

; 1 1

2, 1x, 1x

⎤

⎥⎥

⎦ 3.18

is a special case of2.9fora1,bc1−x.

Thus by evaluating the hypergeometric series by respective summation theorems, we easily obtain the right hand side of the Ramanujan’s summations.

Recently good progress has been done in the direction of generalizing the above- mentioned classical summation theorems 2.2–2.7 see 6. In fact, in a series of three papers by Lavoie et al. 7–9, a large number of very interesting contiguous results of the above mentioned classical summation theorems2.2–2.7 are given. In these papers, the authors have obtained explicit expressions of

2F1

⎡

⎢⎢

⎣

a, b

; −1 1a−bi

⎤

⎥⎥

⎦, 3.19

2F1

⎡

⎢⎢

⎣

a, b

; 1 1 2

2abi1

⎤

⎥⎥

⎦, 3.20

2F1

⎡

⎢⎢

⎢⎣

a, 1−ai

; 1 2 c

⎤

⎥⎥

⎥⎦ 3.21

each fori0,±1,±2,±3,±4,±5, and

3F2

⎡

⎢⎢

⎢⎣

a, b, c

; 1 1

2abi1, 2cj

⎤

⎥⎥

⎥⎦ 3.22

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fori, j0,±1,±2

3F2

⎡

⎢⎣

a, b, c

; 1 1a−bi, 1a−cij

⎤

⎥⎦ 3.23

fori−3,−2,−1,0,1,2;j 0,1,2,3, and

3F2

⎡

⎢⎣ a, b, c

; 1 e, f

⎤

⎥⎦ 3.24

forab1ij,ef 2c1ifori, j0,±1,±2,±3.

Notice that, if we denote3.23byfi,j, the natural symmetry

fi,ja, b, c f_ij,−ja, c, b 3.25 makes it possible to extend the result toj −1,−2,−3.

It is very interesting to mention here that, in order to complete the results3.23of 7×7 matrix, very recently Choi10obtained the remaining ten results.

For i 0, the results 3.19, 3.20, and 3.21 reduce to 2.2, 2.3, and 2.4, respectively, and for i j 0, the results 3.22, 3.23, and3.24 reduce to 2.5,2.6, and2.7, respectively.

On the other hand the following very interesting result for the series₃F2written here in a slightly diﬀerent formis given in the literaturee.g., see11

3F2

⎡

⎢⎣

a, b, d1

; 1 c1, d

⎤

⎥⎦ Γc1Γc−a−b Γc−a1Γc−b1

c−a−b ab d

3.26

providedRc−a−b>0 andRd>0.

Ford c, we get Gauss’s summation theorem2.1. Thus3.26may be regarded as the extension of Gauss’s summation theorem2.1.

Miller12very recently rederived the result3.26and obtained a reduction formula for the Kamp´e de F´eriet function. For comment of Miller’s paper12, see a recent paper by Kim and Rathie13.

The aim of this research paper is to establish the extensions of the above mentioned classical summation theorem 2.2 to 2.9. In the end, as an application, certain very interesting summations, which generalize summations due to Ramanujan have been obtained.

The results are derived with the help of contiguous results of the above mentioned classical summation theorems obtained in a series of three research papers by Lavoie et al.

7–9.

The results derived in this paper are simple, interesting, easily established, and may be useful.

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4. Results Required

The following summation formulas which are special cases of the results 2.2 to 2.7 obtained earlier by Lavoie et al.7–9will be required in our present investigations.

iContiguous Kummer’s theorem9:

2F1

⎡

⎢⎢

⎣

a, b

; −1 2a−b

⎤

⎥⎥

⎦

Γ1/2Γ2a−b 2^a1−b

×

1

Γ1/2a 1/2Γ1/2a−b1− 1

Γ1/2aΓ1/2a−b 3/2

,

2F1

⎡

⎢⎢

⎣

a, b

; −1 3a−b

⎤

⎥⎥

⎦

Γ1/2Γ3a−b 2^a1−b2−b

×

1a−b

Γ1/2a 1/2Γ1/2a−b2− 2

Γ1/2aΓ1/2a−b 3/2

.

4.1

iiContiguous Gauss’s Second theorem9:

2F1

⎡

⎢⎢

⎣

a, b

; 1 1 2

2ab3

⎤

⎥⎥

⎦

Γ1/2Γ1/2a 1/2b 3/2Γ1/2a−1/2b−1/2

Γ1/2a−1/2b 3/2

×

1/2ab−1

Γ1/2a 1/2Γ1/2b 1/2 − 2

Γ1/2aΓ1/2a

.

4.2

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iiiContiguous Bailey’s theorem9:

2F1

⎡

⎢⎢

⎢⎣

a, 3−a

; 1 2 c

⎤

⎥⎥

⎥⎦

Γ1/2ΓcΓ1−a 2^c−3Γ3−a ×

c−2

Γ1/2c−1/2a 1/2Γ1/2c 1/2a−1

− 2

Γ1/2c−1/2aΓ1/2c 1/2a−3/2

. 4.3

ivContiguous Watson’s theorem7:

3F2

⎡

⎢⎢

⎢⎣

a, b, c

; 1 1

2ab1, 2c1

⎤

⎥⎥

⎥⎦

2^ab−2Γc 1/2Γ1/2a 1/2b 1/2Γc−1/2a−1/2b 1/2

Γ1/2ΓaΓb

×

Γ1/2aΓ1/2b

Γc−1/2a 1/2Γc−1/2b 1/2− Γ1/2a 1/2Γ1/2b 1/2

Γc−1/2a1Γc−1/2b1 4.4

provided thatR2c−a−b>−1.

3F2

⎡

⎢⎢

⎢⎣

a, b, c

; 1 1

2ab3, 2c

⎤

⎥⎥

⎥⎦

2^ab1Γc 1/2Γ1/2a 1/2b 3/2Γc−1/2a−1/2b−1/2 a−b−1a−b1Γ1/2ΓaΓb

×

a2c−a b2c−b−2c1 8

Γ1/2aΓ1/2b

Γc−1/2a 1/2Γc−1/2b 1/2

−Γ1/2a 1/2Γ1/2b 1/2

Γc−1/2aΓc−1/2b

4.5

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provided thatR2c−a−b>1.

3F2

⎡

⎢⎢

⎢⎣

a, b, c

; 1 1

2ab3, 2c−1

⎤

⎥⎥

⎥⎦

2^ab−1Γc−1/2Γ1/2a 1/2b 3/2Γc−1/2a−1/2b−1/2 a−b−1a−b1Γ1/2ΓaΓb

×

ab−1Γ1/2aΓ1/2b

Γc−1/2a−1/2Γc−1/2b−1/2

−4c−a−b−3Γ1/2a 1/2Γ1/2b 1/2 Γc−1/2aΓc−1/2b

4.6

provided thatR2c−a−b>1.

vContiguous Dixon’s theorem8:

3F2

⎡

⎢⎢

⎣

a, b, c

; 1 2a−b, 2a−c

⎤

⎥⎥

⎦ 2^−2c1Γ2a−bΓ2a−c b−1c−1Γa−2c2Γa−b−c2

×

Γ1/2a−c 3/2Γ1/2a−b−c2 Γ1/2a 1/2Γ1/2a−b1

−Γ1/2a−c1Γ1/2a−b−c 5/2 Γ1/2aΓ1/2a−b 3/2

4.7

provided thatRa−2b−2c>−4.

3F2

⎡

⎢⎢

⎣

a, b, c

; 1 2a−b, 1a−c

⎤

⎥⎥

⎦ 2^−2b1Γ1a−cΓ2a−b b−1Γa−2b2Γa−b−c2

×

Γ1/2a−b1Γ1/2a−b−c 3/2

Γ1/2aΓ1/2a−c 1/2

−Γ1/2a−b 3/2Γ1/2a−b−c2 Γ1/2a 1/2Γ1/2a−c1

4.8

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3F2

⎡

⎢⎢

⎣

a, b, c

; 1 2a−c, 3a−b

⎤

⎥⎥

⎦

2^−2b2Γ2a−cΓ3a−b

b−1b−2c−1Γa−2b3Γa−b−c3

×

a−2c−b3Γ1/2a−b2Γ1/2a−b−c 5/2

Γ1/2aΓ1/2a−c 3/2

−a−b1Γ1/2a−b 3/2Γ1/2a−b−c3 Γ1/2a 1/2Γ1/2a−c1

4.9

viContiguous Whipple’s theorem9:

3F2

⎡

⎢⎢

⎣

a, 1−a, c

; 1 e, 2c2−e

⎤

⎥⎥

⎦ ΓeΓ2c2−eΓe−c−1 2^2aΓe−aΓe−cΓ2c−e−a2

×

Γ1/2e−1/2a 1/2Γc−1/2e−1/2a1 Γ1/2e 1/2a−1/2Γc−1/2e 1/2a1

−Γ1/2e−1/2aΓc−1/2e−1/2a 3/2

Γ1/2e 1/2aΓc−1/2e 1/2a 1/2

4.10

provided thatRc>0.

3F2

⎡

⎢⎢

⎣

a, 3−a, c

; 1 e, 2c2−e

⎤

⎥⎥

⎦ ΓeΓ2c−e2Γe−c−1

2^2a−2c−1a−1a−2Γe−aΓe−cΓ2c−e−a2

×

2c−eΓ1/2e−1/2a 1/2Γc−1/2e−1/2a1 Γ1/2e 1/2a−3/2Γc−1/2e 1/2a

−e−2Γ1/2e−1/2aΓc−1/2e−1/2a 3/2 Γ1/2e 1/2a−1Γc−1/2e 1/2a−1/2

4.11

provided thatRc>0.

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5. Main Summation Formulas

In this section, the following extensions of the classical summation theorems will be established. In all these theorems we haveRd>0.

iExtension of Kummer’s theorem:

3F2

⎡

⎢⎢

⎣

a, b, d1

; −1 2a−b, d

⎤

⎥⎥

⎦ Γ1/2Γ2a−b

2^a1−b

1a−b/d−1

Γ1/2aΓ1/2a−b 3/2 1−a/d

Γ1/2a 1/2Γ1/2a−b1

. 5.1

iiExtension of Gauss’s second theorem:

3F2

⎡

⎢⎢

⎣

a, b, d1

; 1 1 2

2ab3, d

⎤

⎥⎥

⎦

Γ1/2Γ1/2a 1/2b 3/2Γ1/2a−1/2b−1/2

Γ1/2a−1/2b 3/2

×

1/2ab1−ab/d

Γ1/2a 1/2Γ1/2b 1/2ab1/d−2

Γ1/2aΓ1/2b

.

5.2

iiiExtension of Bailey’s theorem:

3F2

⎡

⎢⎢

⎢⎣

a, 1−a, d1

; 1 2 c1, d

⎤

⎥⎥

⎥⎦

Γ1/2Γc1 2^c

2/d

Γ1/2c 1/2aΓ1/2c−1/2a 1/2

1−c/d

Γ1/2c−1/2a1Γ1/2c 1/2a 1/2

.

5.3

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ivExtension of Watson’s theorem:

First Extension:

4F3

⎡

⎢⎢

⎣

a, b, c, d1

; 1 1

2ab1, 2c1, d

⎤

⎥⎥

⎦

2^ab−2Γc 1/2Γ1/2a 1/2b 1/2Γc−1/2a−1/2b 1/2

Γ1/2ΓaΓb

×

Γ1/2aΓ1/2b

Γc−1/2a 1/2Γc−1/2b 1/2 2c−d/dΓ1/2a 1/2Γ1/2b 1/2

Γ1/2c−1/2a1Γc−1/2b1

5.4

providedR2c−a−b>−1.

Second Extension:

4F3

⎡

⎢⎢

⎣

a, b, c, d1

; 1 1

2ab3, 2c, d

⎤

⎥⎥

⎦

2^ab−2Γc 1/2Γ1/2a 1/2b 3/2Γc−1/2a−1/2b−1/2 a−b−1a−b1Γ1/2ΓaΓb

×

α Γ1/2aΓ1/2b

Γc−1/2a 1/2Γc−1/2b 1/2βΓ1/2a 1/2Γ1/2b 1/2

Γc−1/2aΓc−1/2b

5.5

providedR2c−a−b>1,αandβare given by

αa2c−a b2c−b−2c1−ab

d4c−a−b−1, β8

1

2dab1−1

.

5.6

vExtension of Dixon’s theorem:

4F3

⎡

⎢⎣

a, b, c, d1

; 1 2a−b, 1a−c, d

⎤

⎥⎦

α

b−1 Γ1a−cΓ2a−bΓ3/2 1/2a−b−cΓ1/2

2^aΓ1/2aΓ1/2a−c 1/2Γ2a−b−cΓ1/2a−b 3/2 β

b−1

2^−a−1Γ1/2Γ1a−cΓ1a−bΓ1 1/2a−b−c Γ1/2a 1/2Γ1 1/2a−bΓ1 1/2a−cΓ1a−b−c

5.7

(17)

providedRa−2b−2c>−2,αandβare given by α1− 1

d1a−b, β 1a−b

1a−b−c a

d1a−b−2c−2 1

2a−b−c1

.

5.8

viExtension of Whipple’s theorem:

4F3

⎡

⎢⎣

a, 1−a, c, d1

; 1 e1, 2c−e1, d

⎤

⎥⎦

2^−2aΓe1Γe−cΓ2c−e1 Γe−a1Γe−c1Γ2c−e−a1

×

1−2c−e d

Γ1/2e−1/2a1Γc−1/2e−1/2a 1/2

Γ1/2e 1/2aΓc−1/2e 1/2a 1/2

e

d−1Γ1/2e−1/2a 1/2Γc−1/2e−1/2a1

Γ1/2e 1/2a 1/2Γc−1/2e 1/2a

5.9

providedRc>0.

viiExtension of2.8:

3F2

⎡

⎢⎣

a, b, 1d

; −1 1a−b, d

⎤

⎥⎦

1− a

2d

Γ1a−bΓ1 1/2a Γ1aΓ1 1/2a−b a

2d

Γ1a−bΓ1/2a 1/2 Γ1aΓ1/2a−b 1/2.

5.10

viiiExtension of2.9:

4F3

⎡

⎢⎣

a, b, c, d1

; 1 1a−b, 1a−c, d

⎤

⎥⎦

1− a

2d

Γ1 1/2aΓ1a−bΓ1a−cΓ1 1/2a−b−c Γ1aΓ1a−b−cΓ1 1/2a−bΓ1 1/2a−c a

2d

Γ1/2 1/2aΓ1a−bΓ1a−cΓ1/2 1/2a−b−c Γ1aΓ1a−b−cΓ1/2 1/2a−bΓ1/2 1/2a−c

5.11

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5.1. Derivations

In order to derive5.1, it is just a simple exercise to prove the following relation:

3F2

⎡

⎢⎣

a, b, d1

; −1 2a−b, d

⎤

⎥⎦

2F1

⎡

⎢⎣

a, b

; −1 2a−b

⎤

⎥⎦− ab

d2a−b²F1

⎡

⎢⎣

a1, b1

; −1 3a−b

⎤

⎥⎦.

5.12

Now, it is easy to see that the first and second₂F1 on the right-hand side of 5.12 can be evaluated with the help of contiguous Kummer’s theorems4.1, and after a little simplification, we arrive at the desired result5.1.

In the exactly same manner, the results5.2to5.11can be established with the help of the following relations:

3F2

⎡

⎢⎢

⎣

a, b, d1

; 1 1 2

2ab3, d

⎤

⎥⎥

⎦

2F1

⎡

⎢⎢

⎣

a, b

; 1 1 2

2ab3

⎤

⎥⎥

⎦ ab

dab3²F1

⎡

⎢⎢

⎣

a1, b1

; 1 1 2

2ab5

⎤

⎥⎥

⎦,

3F2

⎡

⎢⎢

⎣

a, 1−a, d1

; 1 c1, d 2

⎤

⎥⎥

⎦

2F1

⎡

⎢⎢

⎣

a, 1−a

; 1 c1 2

⎤

⎥⎥

⎦ a1−a 2d1c²F1

⎡

⎢⎢

⎣

a1, 2−a

; 1 2 c2

⎤

⎥⎥

⎦,

4F3

⎡

⎢⎢

⎢⎣

a, b, c, d1

; 1 1

2ab1, 2c1, d

⎤

⎥⎥

⎥⎦

3F2

⎡

⎢⎢

⎢⎣

a, b, c

; 1 1

2ab1, 2c1

⎤

⎥⎥

⎥⎦ 2abc

d2c1ab1³F2

⎡

⎢⎢

⎢⎣

a1, b1, c1

; 1 1

2ab3, 2c2

⎤

⎥⎥

⎥⎦,

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4F3

⎡

⎢⎢

⎢⎣

a, b, c, d1

; 1 1

2ab3, 2c, d

⎤

⎥⎥

⎥⎦

3F2

⎡

⎢⎢

⎢⎣

a, b, c

; 1 1

2ab3, 2c

⎤

⎥⎥

⎥⎦ ab

dab3³F2

⎡

⎢⎢

⎢⎣

a1, b1, c1

; 1 1

2ab5, 2c1

⎤

⎥⎥

⎥⎦,

4F3

⎡

⎢⎣

a, b, c, d1

; 1 2a−b, 1a−c, d

⎤

⎥⎦

3F2

⎡

⎢⎣

a, b, c

; 1 2a−b, 1a−c

⎤

⎥⎦ abc

d2a−b1a−c³F2

⎡

⎢⎣

a1, b1, c1

; 1 3a−b, 2a−c

⎤

⎥⎦,

4F3

⎡

⎢⎣

a, 1−a, c, d1

; 1 e1, 2c−e1, d

⎤

⎥⎦

3F2

⎡

⎢⎣

a, 1−a, c

; 1 e1, 2c−e1

⎤

⎥⎦ ac1−a

de12c−e1³F2

⎡

⎢⎣

a1, 2−a, c1

; 1 e2, 2c−e2

⎤

⎥⎦,

3F2

⎡

⎢⎣

a, b, d1

; −1 1a−b, d

⎤

⎥⎦

2F1

⎡

⎢⎣

a, b

; −1 1a−b

⎤

⎥⎦− ab

d1a−b²F1

⎡

⎢⎣

a1, b1

; −1 2a−b

⎤

⎥⎦,

4F3

⎡

⎢⎣

a, b, c, d1

; 1 1a−b, 1a−c, d

⎤

⎥⎦

3F2

⎡

⎢⎣

a, b, c

; 1 1a−b, 1a−c

⎤

⎥⎦ abc

d1a−b1a−c³F2

⎡

⎢⎣

a1, b1, c1

; 1 2a−b, 2a−c

⎤

⎥⎦

5.13

and using the results4.2;2.4,4.3;2.5,4.4;4.5,4.6;4.8,4.9;4.10,4.11;2.2, 4.1, and2.6,4.7, respectively.

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5.2. Special Cases

1In5.1, if we taked1a−b, we get Kummer’s theorem2.2.

2In5.2, if we taked 1/2ab1, we get Gauss’s second theorem2.3.

3In5.3, if we takedc, we get Bailey’s theorem2.4.

4In5.4, if we taked2c, we get Watson’s theorem2.5.

5In5.5, if we taked 1/2ab1, we again get Watson’s theorem2.5.

6In5.7, if we taked1a−b, we get Dixon’s theorem2.6.

7In5.9, if we takede, we get Whipple’s theorem2.7.

8In5.10, if we taked 1/2a, we get2.8.

9In5.11, if we taked 1/2a, we get2.9.

6. Generalizations of Summations Due to Ramanujan

In this section, the following summations, which generalize Ramanujan’s summations3.1 to3.11, will be established.

In all the summations, we haved >0.

iForRx>1/2:

1− x−1

x2

d1 d

x−1x−2 x2x3

d2 d

− · · · x1

2dx2x−1{d2x−1 1−x}, 6.1 1−

1 2

₂ d1

2d

1·3

2·4 ₂

d2 3d

− · · ·√ 2π

1 d−1

4 Γ²1/4

1− 1

2d 1

Γ²3/4

. 6.2

iiForRx>0:

1− x−1

x2

d1 d

x−1x−2 x2x3

d2 d

− · · · Γ1/2Γ2x

2x

1x d −1

1

Γ1/2Γ1x

1− 1 d

1

Γx 1/2

,

6.3

11 2

1 2

₂ d1

2d

1 2²

1·3 2·4

₂ d2

3d

− · · ·√ π

1 dΓ²3/4

1− 1

d 8

Γ²1/4

. 6.4

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iiiForRx>0:

1−1 3

x−1 x2

d1 d

1

5

x−1x−2 x2x3

d2 d

− · · ·

1x d −1

x1 2x2x1π

4

2− 1 d

ΓxΓx2 2x1Γ²x 1/2.

6.5

ivForRx>1/4:

1 x−1² x1x2

d1 d

x−1²x−2² x1x2²x3

d2 d

· · ·

1x

d −1

2^2x−1Γx√2Γx 3/2

πΓ2x1 −

√π 8

x1Γ²xΓ2x1 Γ2xΓ²x 1/2

1

d3x−1−4x1

. 6.6

vForRx>1:

1−3x−1 x2

d1 d

5x−1x−2 x2x3

d2 d

· · · 1

4x

Γx2Γx3 Γ²x 1/2

1−1x d

,

11 5

1 2

₂ d1

2d

1 9

1·3 2·4

₂ d2

3d

· · ·

3 4π

1 d−1

π² 3Γ⁴3/4

1− 1

4d

,

11 5

d1 9d

1 2

1

9²

5d2 13d

1·3 2·4

· · ·

5π^3/2 48√

2Γ²3/4 5

4d −1

− 5 48√

2 π^5/2 Γ²3/4

3 8d−3

2

,

1 1

2 3

d1 2d

1·3 2·4

3 d2

3d

· · ·

π Γ⁴3/4 −3

2

1− 1 d

Γ²3/4 π³ .

6.7

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viForRx<2/3:

1n³ 2

d1 d

1 1!

nn1³ 3.4

d2 d

1 2!· · · 1−1/d

n−1 Γ1/2Γ3/2−2n/2

2ⁿΓn/2Γ1/2−n/2Γ3/2−n/2Γ2−n 2⁻ⁿ⁻¹

n−1²

n/d1−2n−2−3nΓ1/2Γ1−3n/2 Γ²1−n/2Γn/2 1/2Γ1−n .

6.8

viiForRx>1/2:

1−3x−1 x2

d1 d

5x−1x−2 x2x3

d2 d

− · · · x 2d

1− 1

2d

Γ1x√ π

2Γx 1/2. 6.9 viiiForRx>1/2:

1x−1² x1²

d1 d

x−1²x−2² x1²x2²

d2 d

· · ·

1− 1 2d

√

πΓ²1xΓ2x−1/2 2Γ2xΓ²x 1/2 1

2d

Γ²1xΓ2x−1 Γ²xΓ2x .

6.10

6.1. Derivations

3F2

⎡

⎣ 1, 1−x, 1d

; 1 2x, d

⎤

⎦ 6.11

which is a special case of extended Gauss’s summation theorem3.26fora1, b1−xand c1x.

3F2

⎡

⎢⎢

⎣ 1 2, 1

2, d1

; −1 2, d

⎤

⎥⎥

⎦ 6.12

which is a special case of extended Kummer’s summation theorem 5.1for a b 1/2.

Similarly the series6.3corresponds to

3F2

⎡

⎢⎢

⎣ 1

2, 1−x, d1

; −1 2x, d

⎤

⎥⎥

⎦ 6.13

which is a special case of extended Kummer’s theorem5.1fora1 andb1−x.

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3F2

⎡

⎢⎢

⎢⎣ 1 2, 1

2, d1

; 1 2, d 2

⎤

⎥⎥

⎥⎦ 6.14

which is a special case of extended Gauss’s second summation theorem5.2forab1/2 or extended Bailey’s summation theorem5.3fora1/2,c1.

Also, it can be easily seen that the series6.5to6.8which correspond to

4F3

⎡

⎢⎢

⎢⎣ 1, 1

2, 1−x, 1d

; 1 3

2, 2x, d

⎤

⎥⎥

⎥⎦,

4F3

⎡

⎢⎣

1, 1−x, 1x, 1d

; 1 1x, 2x, d

⎤

⎥⎦,

4F3

⎡

⎢⎢

⎢⎣ 1, 3

2, 1−x, 1d

; 1 1

2, 1x, d

⎤

⎥⎥

⎥⎦,

4F3

⎡

⎢⎢

⎢⎣ 1 2, 1

2, 1 4, 1d

; 1 2, 5

4, d

⎤

⎥⎥

⎥⎦,

4F3

⎡

⎢⎢

⎢⎣ 1 2, 1

4, 1 4, 1d

; 1 5

4, 5 4, d

⎤

⎥⎥

⎥⎦,

4F3

⎡

⎢⎢

⎣ 1 2, 1

2, 1 2, 1d

; 1 2, 1, d

⎤

⎥⎥

⎦,

4F3

⎡

⎢⎣

n, n, n, 1d

; 1 1, 2, d

⎤

⎥⎦

6.15

are special cases of extended Dixon’s theorem5.7.

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3F2

⎡

⎢⎢

⎣

1, 1−x, d1

; −1 1x, d

⎤

⎥⎥

⎦ 6.16

which is a special case of5.10fora1 andb1−x. And the series6.10corresponds to

4F3

⎡

⎢⎢

⎣

1, 1−x, 1−x, 1d

; 1 1x, 1x, d

⎤

⎥⎥

⎦ 6.17

which is a special case of5.11fora1,b1−xc.

7. Concluding Remarks

1Various other applications of these results are under investigations and will be published later.

2Further generalizations of the extended summation theorem 5.1to 5.9in the forms

3F2

⎡

⎢⎢

⎣

a, b, d1

; −1 2a−bi, d

⎤

⎥⎥

⎦,

3F2

⎡

⎢⎢

⎣

a, b, d1

; 1 2 1

2ab3i, d

⎤

⎥⎥

⎦ ,

3F2

⎡

⎢⎢

⎣

a, 1−ai, d1

; 1 c1, d

⎤

⎥⎥

⎦

7.1

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each fori0,±1,±2,±3,±4,±5, and

4F3

⎡

⎢⎢

⎢⎣

a, b, c, d1

; 1 1

2abi1, 2cj, d

⎤

⎥⎥

⎥⎦,

4F3

⎡

⎢⎢

⎣

a, b, c, d1

; 1 2a−bi, 1a−cij, d

⎤

⎥⎥

⎦,

7.2

each fori, j 0,±1,±2,±3, and

4F3

⎡

⎢⎢

⎣

a, b, c, 1d

; 1 e, f, d

⎤

⎥⎥

⎦, 7.3

whereab1ij,ef 2cj fori, j 0,±1,±2,±3 are also under investigations and will be published later.

Acknowledgments

The authors are highly grateful to the referees for carefully reading the manuscript and providing certain very useful suggestions which led to a better presentation of this research article. They are so much appreciated to the College of Science, Sultan Qaboos University, Muscat - Oman for supporting the publication charges of this paper. The first author is supported by the Research Fund of Wonkwang University 2011 and the second author is supported by the research grant IG/SCI/DOMS/10/03 of Sultan Qaboos University, OMAN.

References

1 C. F. Gauss, “Disquisitiones generales circa seriem infinitamαβ/1.γx αα1ββ1/1.2.γγ 1x²αα1α2ββ1β2/1.2.3.γγ1γ2x³etc. pars prior,” Commentationes Societiones Regiae Scientiarum Gottingensis Recentiores, vol. 2, 1812, reprinted in Gesammelte Werke, vol. 3, pp.

123–163 and 207–229, 1866.

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