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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2010Β (2010), Article IDΒ 309503, 27 pages
http://dx.doi.org/10.1155/2010/309503
Research Article

Extensions of Certain Classical Summation Theorems for the Series 2𝐹1, 3𝐹2, and 4𝐹3 with Applications in Ramanujan's Summations

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

Received 20 May 2010; Revised 7 September 2010; Accepted 23 September 2010

Academic Editor: Teodor Bulboacă

Copyright Β© 2010 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.

Abstract

Motivated by the extension of classical Gauss's summation theorem for the series 2𝐹1 given 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 series 2𝐹1, Watson, Dixon and Whipple for the series 3𝐹2, and a few other hypergeometric identities for the series 3𝐹2 and 4𝐹3. 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, Gauss [1] systematically discussed the seriesξ“βˆžπ‘›=0(π‘Ž)𝑛(𝑏)𝑛(𝑐)𝑛𝑧𝑛𝑛!=1+π‘Žβ‹…π‘1⋅𝑐𝑧+π‘Ž(π‘Ž+1)𝑏(𝑏+1)𝑧1β‹…2𝑐(𝑐+1)2+β‹―,(1.1) where (πœ†)𝑛 denotes the Pochhammer symbol defined (for πœ†βˆˆβ„‚) by(πœ†)π‘›ξƒ―βˆΆ=1(𝑛=0)πœ†(πœ†+1)β‹―(πœ†+π‘›βˆ’1)(π‘›βˆˆβ„•βˆΆ={1,2,3,…}).(1.2) It is noted that the series (1.1) and its natural generalization π‘ƒπΉπ‘ž in (1.6) are of great importance to mathematicians and physicists. This series (1.1) has been known as the Gauss series or the ordinary hypergeometric series and may be regarded as a generalization of the elementary geometric series. In fact (1.1) reduces to the geometric series in two cases, when π‘Ž=𝑐 and 𝑏=1 also when 𝑏=𝑐 and π‘Ž=1. The series (1.1) is represented by the notation 2𝐹1[π‘Ž,𝑏;𝑐;𝑧] or2𝐹1βŽ‘βŽ’βŽ’βŽ£π‘βŽ€βŽ₯βŽ₯⎦,π‘Ž,𝑏;𝑧(1.3) which is usually referred to as Gauss hypergeometric function. In (1.1), the three elements π‘Ž, 𝑏, and 𝑐 are described as the parameters of the series, and 𝑧 is called the variable of the series. All four of these quantities may be real or complex with an exception that 𝑐 is neither zero nor a negative integer. Also, in (1.1), it is easy to see that if any one of the numerator parameters π‘Ž or 𝑏 or both is a negative integer, then the series reduces to a polynomials, that is, the series terminates.

The series (1.1) is absolutely convergent within the unit circle when |𝑧|<1 provided that 𝑐≠0,βˆ’1,βˆ’2,…. Also when |𝑧|=1, the series is absolutely convergent if β„œ(π‘βˆ’π‘Žβˆ’π‘)>0, conditionally convergent if βˆ’1<β„œ(π‘βˆ’π‘Žβˆ’π‘)≀0, 𝑧≠1 and divergent if β„œ(π‘βˆ’π‘Žβˆ’π‘)β‰€βˆ’1.

Further, if in (1.1), we replace 𝑧 by 𝑧/𝑏 and let π‘β†’βˆž, then ((𝑏)𝑛𝑧𝑛/𝑏𝑛)→𝑧𝑛, and we arrive to the following Kummer's seriesβˆžξ“π‘›=0(π‘Ž)𝑛(𝑐)π‘›π‘§π‘›π‘Žπ‘›!=1+1⋅𝑐𝑧+π‘Ž(π‘Ž+1)𝑧1β‹…2𝑐(𝑐+1)2+β‹―.(1.4)

This series is absolutely convergent for all values of π‘Ž, 𝑐, and 𝑧, real or complex, excluding 𝑐=0,βˆ’1,βˆ’2,… and is represented by the notation 1𝐹1(π‘Ž;𝑐;𝑧) or1𝐹1βŽ‘βŽ’βŽ’βŽ£π‘βŽ€βŽ₯βŽ₯⎦,π‘Ž,;𝑧(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. Thus 2𝐹1 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. Erdélyi, and H. Exton.

A natural generalization of 2𝐹1 is the generalized hypergeometric series π‘πΉπ‘ž defined byπ‘πΉπ‘žβŽ‘βŽ’βŽ’βŽ£π‘Ž1β‹―π‘Žπ‘π‘;𝑧1β‹―π‘π‘žβŽ€βŽ₯βŽ₯⎦=ξ“βˆžπ‘›=0ξ€·π‘Ž1ξ€Έπ‘›β‹―ξ€·π‘Žπ‘ξ€Έπ‘›ξ€·π‘1ξ€Έπ‘›β‹―ξ€·π‘π‘žξ€Έπ‘›π‘§π‘›.𝑛!(1.6)

The series (1.6) is convergent for all |𝑧|<∞ if π‘β‰€π‘ž and for |𝑧|<1 if 𝑝=π‘ž+1 while it is divergent for all 𝑧, 𝑧≠0 if 𝑝>π‘ž+1. When |𝑧|=1 with 𝑝=π‘ž+1, the series (1.6) converges absolutely ifβ„œξƒ©π‘žξ“π‘—=1π‘π‘—βˆ’π‘ξ“π‘—=1π‘Žπ‘—ξƒͺ>0,(1.7) conditionally convergent ifξƒ©βˆ’1<β„œπ‘žξ“π‘—=1π‘π‘—βˆ’π‘ξ“π‘—=1π‘Žπ‘—ξƒͺ≀0,𝑧≠1(1.8) and divergent ifβ„œξƒ©π‘žξ“π‘—=1π‘π‘—βˆ’π‘ξ“π‘—=1π‘Žπ‘—ξƒͺβ‰€βˆ’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 Berndt [2], 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 Bailey [3], a large number of very interesting results involving products of generalized hypergeometric series have been developed. In [4] a 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.

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 series 2𝐹1 and Watson, Dixon, and Whipple for the series 3𝐹2 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 series 2𝐹1 and 3𝐹2. These are the following.

Gauss theorem [5]:2𝐹1βŽ‘βŽ’βŽ’βŽ£π‘βŽ€βŽ₯βŽ₯⎦=π‘Ž,𝑏;1Ξ“(𝑐)Ξ“(π‘βˆ’π‘Žβˆ’π‘)Ξ“(π‘βˆ’π‘Ž)Ξ“(π‘βˆ’π‘)(2.1) provided β„œ(π‘βˆ’π‘Žβˆ’π‘)>0.

Kummer theorem [5]:2𝐹1⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,𝑏;βˆ’11+π‘Žβˆ’π‘Ξ“(1+π‘Žβˆ’π‘)Ξ“(1+(1/2)π‘Ž).Ξ“(1+(1/2)π‘Žβˆ’π‘)Ξ“(1+π‘Ž)(2.2)

Gauss’s second theorem [5]:2𝐹1⎑⎒⎒⎒⎒⎣;1π‘Ž,𝑏212⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=(π‘Ž+𝑏+1)Ξ“(1/2)Ξ“((1/2)π‘Ž+(1/2)𝑏+(1/2)).Ξ“((1/2)π‘Ž+(1/2))Ξ“((1/2)𝑏+(1/2))(2.3)

Bailey theorem [5]:2𝐹1⎑⎒⎒⎒⎣;1π‘Ž,1βˆ’π‘Ž2π‘βŽ€βŽ₯βŽ₯βŽ₯⎦=Ξ“((1/2)𝑐)Ξ“((1/2)𝑐+(1/2)).Ξ“((1/2)𝑐+(1/2)π‘Ž)Ξ“((1/2)π‘βˆ’(1/2)π‘Ž+(1/2))(2.4)

Watson theorem [5]:3𝐹2⎑⎒⎒⎒⎣1π‘Ž,𝑏,𝑐;12⎀βŽ₯βŽ₯βŽ₯⎦=(π‘Ž+𝑏+1),2𝑐Γ(1/2)Ξ“(𝑐+(1/2))Ξ“((1/2)π‘Ž+(1/2)𝑏+(1/2))Ξ“(π‘βˆ’(1/2)π‘Žβˆ’(1/2)𝑏+(1/2))Ξ“((1/2)π‘Ž+(1/2))Ξ“((1/2)𝑏+(1/2))Ξ“(π‘βˆ’(1/2)π‘Ž+(1/2))Ξ“(π‘βˆ’(1/2)𝑏+(1/2))(2.5) provided β„œ(2π‘βˆ’π‘Žβˆ’π‘)>βˆ’1.

Dixon theorem [5]:3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,𝑏,𝑐;11+π‘Žβˆ’π‘,1+π‘Žβˆ’π‘Ξ“(1+(1/2)π‘Ž)Ξ“(1+π‘Žβˆ’π‘)Ξ“(1+π‘Žβˆ’π‘)Ξ“(1+(1/2)π‘Žβˆ’π‘βˆ’π‘)Ξ“(1+π‘Ž)Ξ“(1+(1/2)π‘Žβˆ’π‘)Ξ“(1+(1/2)π‘Žβˆ’π‘)Ξ“(1+π‘Žβˆ’π‘βˆ’π‘)(2.6) provided β„œ(π‘Žβˆ’2π‘βˆ’2𝑐)>βˆ’2.

Whipple theorem [5]:3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,𝑏,𝑐;1𝑒,π‘“πœ‹Ξ“(𝑒)Ξ“(𝑓)22π‘βˆ’1Ξ“((1/2)π‘Ž+(1/2)𝑒)Ξ“((1/2)π‘Ž+(1/2)𝑓)Ξ“((1/2)𝑏+(1/2)𝑒)Ξ“((1/2)𝑏+(1/2)𝑓)(2.7) provided β„œ(𝑐)>0 and β„œ(𝑒+π‘“βˆ’π‘Žβˆ’π‘βˆ’π‘)>0 with π‘Ž+𝑏=1 and 𝑒+𝑓=2𝑐+1.

Other hypergeometric identities [5]:3𝐹2⎑⎒⎒⎒⎒⎣1π‘Ž,1+21π‘Ž,𝑏;12⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=π‘Ž,1+π‘Žβˆ’π‘Ξ“(1+π‘Žβˆ’π‘)Ξ“((1/2)π‘Ž+(1/2)),Ξ“(1+π‘Ž)Ξ“((1/2)π‘Žβˆ’π‘+(1/2))(2.8)4𝐹3⎑⎒⎒⎒⎒⎣1π‘Ž,1+21π‘Ž,𝑏,𝑐;12⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=Ξ“π‘Ž,1+π‘Žβˆ’π‘,1+π‘Žβˆ’π‘(1+π‘Žβˆ’π‘)Ξ“(1+π‘Žβˆ’π‘)Ξ“((1/2)π‘Ž+(1/2))Ξ“((1/2)π‘Žβˆ’π‘βˆ’π‘+(1/2))Ξ“(1+π‘Ž)Ξ“(1+π‘Žβˆ’π‘βˆ’π‘)Ξ“((1/2)π‘Žβˆ’π‘+(1/2))Ξ“((1/2)π‘Žβˆ’π‘+(1/2))(2.9) provided β„œ(π‘Žβˆ’2π‘βˆ’2𝑐)>βˆ’1.

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, SaalschΓΌtz, 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.

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 Berndt [2] that 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 Ramanujan [2].(i)For β„œ(π‘₯)>1/2,1βˆ’(π‘₯βˆ’1)(+π‘₯+1)(π‘₯βˆ’1)(π‘₯βˆ’2)(π‘₯π‘₯+1)(π‘₯+2)βˆ’β‹―=,2π‘₯βˆ’1(3.1)ξ‚€11βˆ’22+ξ‚€1β‹…32β‹…42βˆ’ξ‚€1β‹…3β‹…52β‹…4β‹…62√+β‹―=πœ‹βˆš2Ξ“2.(3/4)(3.2)(ii)For β„œ(π‘₯)>0,1+(π‘₯βˆ’1)+(π‘₯+1)(π‘₯βˆ’1)(π‘₯βˆ’2)2(π‘₯+1)(π‘₯+2)+β‹―=2π‘₯βˆ’1Ξ“2(π‘₯+1),Ξ“(2π‘₯+1)(3.3)11+2ξ‚€122+122ξ‚€1β‹…32β‹…42+123ξ‚€1β‹…3β‹…52β‹…4β‹…62√+β‹―=πœ‹Ξ“2.(3/4)(3.4)(iii)For β„œ(π‘₯)>0,11βˆ’3(π‘₯βˆ’1)+1(π‘₯+1)5(π‘₯βˆ’1)(π‘₯βˆ’2)2(π‘₯+1)(π‘₯+2)βˆ’β‹―=4π‘₯Ξ“4(π‘₯+1)4π‘₯Ξ“2(.2π‘₯+1)(3.5)(iv)For β„œ(π‘₯)>1/4,1+(π‘₯βˆ’1)2(π‘₯+1)2+ξ‚Έ(π‘₯βˆ’1)(π‘₯βˆ’2)(ξ‚Ήπ‘₯+1)(π‘₯+2)2+β‹―=2π‘₯Ξ“4π‘₯βˆ’14(π‘₯+1)Ξ“(4π‘₯+1)Ξ“4.(2π‘₯+1)(3.6)(v)For β„œ(π‘₯)>1,1βˆ’3(π‘₯βˆ’1)(π‘₯+1)+5(π‘₯βˆ’1)(π‘₯βˆ’2)(π‘₯+1)(π‘₯+2)βˆ’β‹―=0,(3.7)11+5ξ‚€122+19ξ‚€1β‹…32β‹…42πœ‹+β‹―=24Ξ“4,1(3/4)1+52ξ‚€12+192ξ‚€1β‹…3ξ‚πœ‹2β‹…4+β‹―=5/28√2Ξ“2,ξ‚€1(3/4)1+22+ξ‚€1β‹…32β‹…42πœ‹+β‹―=Ξ“4.(3/4)(3.8)(vi)For β„œ(π‘₯)<2/3,ξ‚€π‘₯1+1!3+ξ‚΅π‘₯(π‘₯+1)ξ‚Ά2!3+β‹―=6sin(πœ‹π‘₯/2)sin(πœ‹π‘₯)Ξ“3((1/2)π‘₯+1)πœ‹2π‘₯2Ξ“.((3/2)π‘₯+1)(1+2cosπœ‹π‘₯)(3.9)(vii)For β„œ(π‘₯)>1/2,1+3(π‘₯βˆ’1)(π‘₯+1)+5(π‘₯βˆ’1)(π‘₯βˆ’2)(π‘₯+1)(π‘₯+2)+β‹―=π‘₯.(3.10)(viii)For β„œ(π‘₯)>1/2,ξ‚Έ1+3(π‘₯βˆ’1)(ξ‚Ήπ‘₯+1)2ξ‚Έ+5(π‘₯βˆ’1)(π‘₯βˆ’2)(ξ‚Ήπ‘₯+1)(π‘₯+2)2π‘₯+β‹―=2.2π‘₯βˆ’1(3.11) We now come to the derivations of these summation in brief.

It is easy to see that the series (3.1) corresponds to2𝐹1⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦1,1βˆ’π‘₯;11+π‘₯(3.12) which is a special case of Gauss's summation theorem (2.1) for π‘Ž=1, 𝑏=1βˆ’π‘₯ and 𝑐=1+π‘₯.

The series (3.2) corresponds to2𝐹1⎑⎒⎒⎒⎣12,121⎀βŽ₯βŽ₯βŽ₯⎦;βˆ’1(3.13) which is a special case of Kummer's summation theorem (2.2) for π‘Ž=𝑏=1/2. Similarly the series (3.3) corresponds to2𝐹1⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦1,1βˆ’π‘₯;βˆ’11+π‘₯(3.14) which is a special case of Kummer's summation theorem (2.2) for π‘Ž=1, 𝑏=1βˆ’π‘₯.

The series (3.4) corresponds to2𝐹1⎑⎒⎒⎒⎣12,12;121⎀βŽ₯βŽ₯βŽ₯⎦(3.15) which is a special case of Gauss's second summation theorem (2.3) for π‘Ž=𝑏=1/2 or Bailey's summation theorem (2.4) for π‘Ž=1/2 and 𝑐=1.

Also, it can easily be seen that the series (3.5) to (3.9) correspond to each of the following series: 3𝐹2⎑⎒⎒⎒⎒⎣11,23,1βˆ’π‘₯;12⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,,1+π‘₯3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,1,1βˆ’π‘₯,1βˆ’π‘₯;11+π‘₯,1+π‘₯3𝐹2⎑⎒⎒⎒⎒⎣31,21,1βˆ’π‘₯;12⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,,1+π‘₯3𝐹2⎑⎒⎒⎒⎒⎣12,12,145;11,4⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,3𝐹2⎑⎒⎒⎒⎒⎣12,14,145;14,54⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,3𝐹2⎑⎒⎒⎒⎣12,12,12⎀βŽ₯βŽ₯βŽ₯⎦,;11,13𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,π‘₯,π‘₯,π‘₯;11,1(3.16) which are special cases of classical Dixon's theorem (2.6) for (i) π‘Ž=1, 𝑏=1/2, 𝑐=1βˆ’π‘₯, (ii) π‘Ž=1, 𝑏=𝑐=1βˆ’π‘₯, (iii) π‘Ž=1, 𝑏=3/2, 𝑐=1βˆ’π‘₯, (iv) π‘Ž=𝑏=1/2, 𝑐=1/4, (v) π‘Ž=1/2, 𝑏=𝑐=1/4, (vi) π‘Ž=𝑏=𝑐=1/2, and (vii) π‘Ž=𝑏=𝑐=π‘₯, respectively.

The series (3.10) which corresponds to3𝐹2⎑⎒⎒⎒⎒⎣31,21,1βˆ’π‘₯;βˆ’12⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,1+π‘₯(3.17) is a special case of (2.8) for π‘Ž=1, 𝑏=1βˆ’π‘₯, and the series (3.11) which corresponds to4𝐹3⎑⎒⎒⎒⎒⎣31,21,1βˆ’π‘₯,1βˆ’π‘₯;12⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,1+π‘₯,1+π‘₯(3.18) is a special case of (2.9) for π‘Ž=1, 𝑏=𝑐=1βˆ’π‘₯.

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 theorems (2.2)–(2.7) are given. In these papers, the authors have obtained explicit expressions of2𝐹1⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,π‘Ž,𝑏;βˆ’11+π‘Žβˆ’π‘+𝑖(3.19)2𝐹1⎑⎒⎒⎒⎒⎣;1π‘Ž,𝑏212⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,(π‘Ž+𝑏+𝑖+1)(3.20)2𝐹1⎑⎒⎒⎒⎣;1π‘Ž,1βˆ’π‘Ž+𝑖2π‘βŽ€βŽ₯βŽ₯βŽ₯⎦(3.21) each for 𝑖=0,Β±1,Β±2,Β±3,Β±4,Β±5, and3𝐹2⎑⎒⎒⎒⎣1π‘Ž,𝑏,𝑐;12⎀βŽ₯βŽ₯βŽ₯⎦(π‘Ž+𝑏+𝑖+1),2𝑐+𝑗(3.22) for 𝑖,𝑗=0,Β±1,Β±23𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦π‘Ž,𝑏,𝑐;11+π‘Žβˆ’π‘+𝑖,1+π‘Žβˆ’π‘+𝑖+𝑗(3.23) for 𝑖=βˆ’3,βˆ’2,βˆ’1,0,1,2; 𝑗=0,1,2,3, and3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦π‘Ž,𝑏,𝑐;1𝑒,𝑓(3.24) for π‘Ž+𝑏=1+𝑖+𝑗, 𝑒+𝑓=2𝑐+1+𝑖 for 𝑖,𝑗=0,Β±1,Β±2,Β±3.

Notice that, if we denote (3.23) by 𝑓𝑖,𝑗, the natural symmetry𝑓𝑖,𝑗(π‘Ž,𝑏,𝑐)=𝑓𝑖+𝑗,βˆ’π‘—(π‘Ž,𝑐,𝑏)(3.25) makes it possible to extend the result to 𝑗=βˆ’1,βˆ’2,βˆ’3.

It is very interesting to mention here that, in order to complete the results (3.23) of 7Γ—7 matrix, very recently Choi [10] obtained 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), and (3.24) reduce to (2.5), (2.6), and (2.7), respectively.

On the other hand the following very interesting result for the series 3𝐹2 (written here in a slightly different form) is given in the literature (e.g., see [11])3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,𝑏,𝑑+1;1𝑐+1,𝑑Γ(𝑐+1)Ξ“(π‘βˆ’π‘Žβˆ’π‘)Γ(π‘βˆ’π‘Ž+1)Ξ“(π‘βˆ’π‘+1)(π‘βˆ’π‘Žβˆ’π‘)+π‘Žπ‘π‘‘ξ‚„(3.26) provided β„œ(π‘βˆ’π‘Žβˆ’π‘)>0 and β„œ(𝑑)>0.

For 𝑑=𝑐, we get Gauss's summation theorem (2.1). Thus (3.26) may be regarded as the extension of Gauss's summation theorem (2.1).

Miller [12] very recently rederived the result (3.26) and obtained a reduction formula for the KampΓ© de FΓ©riet function. For comment of Miller's paper [12], see a recent paper by Kim and Rathie [13].

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.

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. [7–9] will be required in our present investigations.(i)Contiguous Kummer's theorem [9]:2𝐹1⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,𝑏;βˆ’12+π‘Žβˆ’π‘Ξ“(1/2)Ξ“(2+π‘Žβˆ’π‘)2π‘ŽΓ—ξ‚Έ1(1βˆ’π‘)βˆ’1Ξ“((1/2)π‘Ž+(1/2))Ξ“((1/2)π‘Žβˆ’π‘+1)ξ‚Ή,Ξ“((1/2)π‘Ž)Ξ“((1/2)π‘Žβˆ’π‘+(3/2))2𝐹1⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,𝑏;βˆ’13+π‘Žβˆ’π‘Ξ“(1/2)Ξ“(3+π‘Žβˆ’π‘)2π‘ŽΓ—ξ‚Έ(1βˆ’π‘)(2βˆ’π‘)(1+π‘Žβˆ’π‘)Ξ“βˆ’2((1/2)π‘Ž+(1/2))Ξ“((1/2)π‘Žβˆ’π‘+2)Ξ“ξ‚Ή.((1/2)π‘Ž)Ξ“((1/2)π‘Žβˆ’π‘+(3/2))(4.1)(ii)Contiguous Gauss's Second theorem [9]:2𝐹1⎑⎒⎒⎒⎒⎣;1π‘Ž,𝑏212⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=(π‘Ž+𝑏+3)Ξ“(1/2)Ξ“((1/2)π‘Ž+(1/2)𝑏+(3/2))Ξ“((1/2)π‘Žβˆ’(1/2)π‘βˆ’(1/2))Γ—ξ‚ΈΞ“((1/2)π‘Žβˆ’(1/2)𝑏+(3/2))(1/2)(π‘Ž+π‘βˆ’1)βˆ’2Ξ“((1/2)π‘Ž+(1/2))Ξ“((1/2)𝑏+(1/2))ξ‚Ή.Ξ“((1/2)π‘Ž)Ξ“((1/2)π‘Ž)(4.2)(iii)Contiguous Bailey's theorem [9]:2𝐹1⎑⎒⎒⎒⎣;1π‘Ž,3βˆ’π‘Ž2π‘βŽ€βŽ₯βŽ₯βŽ₯⎦=Ξ“(1/2)Ξ“(𝑐)Ξ“(1βˆ’π‘Ž)2π‘βˆ’3Γ—ξ‚ΈΞ“(3βˆ’π‘Ž)(π‘βˆ’2)βˆ’2Ξ“((1/2)π‘βˆ’(1/2)π‘Ž+(1/2))Ξ“((1/2)𝑐+(1/2)π‘Žβˆ’1)ξ‚Ή.Ξ“((1/2)π‘βˆ’(1/2)π‘Ž)Ξ“((1/2)𝑐+(1/2)π‘Žβˆ’(3/2))(4.3)(iv)Contiguous Watson's theorem [7]: 3𝐹2⎑⎒⎒⎒⎣1π‘Ž,𝑏,𝑐;12⎀βŽ₯βŽ₯βŽ₯⎦=2(π‘Ž+𝑏+1),2𝑐+1π‘Ž+π‘βˆ’2Ξ“(𝑐+(1/2))Ξ“((1/2)π‘Ž+(1/2)𝑏+(1/2))Ξ“(π‘βˆ’(1/2)π‘Žβˆ’(1/2)𝑏+(1/2))Γ—ξ‚ΈΞ“(1/2)Ξ“(π‘Ž)Ξ“(𝑏)Ξ“((1/2)π‘Ž)Ξ“((1/2)𝑏)βˆ’Ξ“(π‘βˆ’(1/2)π‘Ž+(1/2))Ξ“(π‘βˆ’(1/2)𝑏+(1/2))Ξ“((1/2)π‘Ž+(1/2))Ξ“((1/2)𝑏+(1/2))ξ‚ΉΞ“(π‘βˆ’(1/2)π‘Ž+1)Ξ“(π‘βˆ’(1/2)𝑏+1)(4.4) provided that β„œ(2π‘βˆ’π‘Žβˆ’π‘)>βˆ’1.3𝐹2⎑⎒⎒⎒⎣1π‘Ž,𝑏,𝑐;12⎀βŽ₯βŽ₯βŽ₯⎦=2(π‘Ž+𝑏+3),2π‘π‘Ž+𝑏+1Ξ“(𝑐+(1/2))Ξ“((1/2)π‘Ž+(1/2)𝑏+(3/2))Ξ“(π‘βˆ’(1/2)π‘Žβˆ’(1/2)π‘βˆ’(1/2))(Γ—ξ‚Έπ‘Žβˆ’π‘βˆ’1)(π‘Žβˆ’π‘+1)Ξ“(1/2)Ξ“(π‘Ž)Ξ“(𝑏)π‘Ž(2π‘βˆ’π‘Ž)+𝑏(2π‘βˆ’π‘)βˆ’2𝑐+18Ξ“((1/2)π‘Ž)Ξ“((1/2)𝑏)βˆ’Ξ“(π‘βˆ’(1/2)π‘Ž+(1/2))Ξ“(π‘βˆ’(1/2)𝑏+(1/2))Ξ“((1/2)π‘Ž+(1/2))Ξ“((1/2)𝑏+(1/2))Ξ“ξ‚Ή(π‘βˆ’(1/2)π‘Ž)Ξ“(π‘βˆ’(1/2)𝑏)(4.5) provided that β„œ(2π‘βˆ’π‘Žβˆ’π‘)>1.3𝐹2⎑⎒⎒⎒⎣1π‘Ž,𝑏,𝑐;12⎀βŽ₯βŽ₯βŽ₯⎦=2(π‘Ž+𝑏+3),2π‘βˆ’1π‘Ž+π‘βˆ’1Ξ“(π‘βˆ’(1/2))Ξ“((1/2)π‘Ž+(1/2)𝑏+(3/2))Ξ“(π‘βˆ’(1/2)π‘Žβˆ’(1/2)π‘βˆ’(1/2))(Γ—ξ‚Έπ‘Žβˆ’π‘βˆ’1)(π‘Žβˆ’π‘+1)Ξ“(1/2)Ξ“(π‘Ž)Ξ“(𝑏)(π‘Ž+π‘βˆ’1)Ξ“((1/2)π‘Ž)Ξ“((1/2)𝑏)βˆ’Ξ“(π‘βˆ’(1/2)π‘Žβˆ’(1/2))Ξ“(π‘βˆ’(1/2)π‘βˆ’(1/2))(4π‘βˆ’π‘Žβˆ’π‘βˆ’3)Ξ“((1/2)π‘Ž+(1/2))Ξ“((1/2)𝑏+(1/2))Ξ“ξ‚Ή(π‘βˆ’(1/2)π‘Ž)Ξ“(π‘βˆ’(1/2)𝑏)(4.6) provided that β„œ(2π‘βˆ’π‘Žβˆ’π‘)>1.(v)Contiguous Dixon's theorem [8]:3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=2π‘Ž,𝑏,𝑐;12+π‘Žβˆ’π‘,2+π‘Žβˆ’π‘βˆ’2𝑐+1Ξ“(2+π‘Žβˆ’π‘)Ξ“(2+π‘Žβˆ’π‘)Γ—ξ‚Έ(π‘βˆ’1)(π‘βˆ’1)Ξ“(π‘Žβˆ’2𝑐+2)Ξ“(π‘Žβˆ’π‘βˆ’π‘+2)Ξ“((1/2)π‘Žβˆ’π‘+(3/2))Ξ“((1/2)π‘Žβˆ’π‘βˆ’π‘+2)Ξ“βˆ’Ξ“((1/2)π‘Ž+(1/2))Ξ“((1/2)π‘Žβˆ’π‘+1)((1/2)π‘Žβˆ’π‘+1)Ξ“((1/2)π‘Žβˆ’π‘βˆ’π‘+(5/2))ξ‚ΉΞ“((1/2)π‘Ž)Ξ“((1/2)π‘Žβˆ’π‘+(3/2))(4.7) provided that β„œ(π‘Žβˆ’2π‘βˆ’2𝑐)>βˆ’4.3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=2π‘Ž,𝑏,𝑐;12+π‘Žβˆ’π‘,1+π‘Žβˆ’π‘βˆ’2𝑏+1Ξ“(1+π‘Žβˆ’π‘)Ξ“(2+π‘Žβˆ’π‘)Γ—ξ‚Έ(π‘βˆ’1)Ξ“(π‘Žβˆ’2𝑏+2)Ξ“(π‘Žβˆ’π‘βˆ’π‘+2)Ξ“((1/2)π‘Žβˆ’π‘+1)Ξ“((1/2)π‘Žβˆ’π‘βˆ’π‘+(3/2))Ξ“βˆ’Ξ“((1/2)π‘Ž)Ξ“((1/2)π‘Žβˆ’π‘+(1/2))((1/2)π‘Žβˆ’π‘+(3/2))Ξ“((1/2)π‘Žβˆ’π‘βˆ’π‘+2)ξ‚ΉΞ“((1/2)π‘Ž+(1/2))Ξ“((1/2)π‘Žβˆ’π‘+1)(4.8) provided that β„œ(π‘Žβˆ’2π‘βˆ’2𝑐)>βˆ’3.3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=2π‘Ž,𝑏,𝑐;12+π‘Žβˆ’π‘,3+π‘Žβˆ’π‘βˆ’2𝑏+2Ξ“(2+π‘Žβˆ’π‘)Ξ“(3+π‘Žβˆ’π‘)Γ—ξ‚Έ(π‘βˆ’1)(π‘βˆ’2)(π‘βˆ’1)Ξ“(π‘Žβˆ’2𝑏+3)Ξ“(π‘Žβˆ’π‘βˆ’π‘+3)(π‘Žβˆ’2π‘βˆ’π‘+3)Ξ“((1/2)π‘Žβˆ’π‘+2)Ξ“((1/2)π‘Žβˆ’π‘βˆ’π‘+(5/2))βˆ’Ξ“((1/2)π‘Ž)Ξ“((1/2)π‘Žβˆ’π‘+(3/2))(π‘Žβˆ’π‘+1)Ξ“((1/2)π‘Žβˆ’π‘+(3/2))Ξ“((1/2)π‘Žβˆ’π‘βˆ’π‘+3)ξ‚ΉΞ“((1/2)π‘Ž+(1/2))Ξ“((1/2)π‘Žβˆ’π‘+1)(4.9) provided that β„œ(π‘Žβˆ’2π‘βˆ’2𝑐)>βˆ’3.(vi)Contiguous Whipple's theorem [9]:3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,1βˆ’π‘Ž,𝑐;1𝑒,2𝑐+2βˆ’π‘’Ξ“(𝑒)Ξ“(2𝑐+2βˆ’π‘’)Ξ“(π‘’βˆ’π‘βˆ’1)22π‘ŽΓ—ξ‚ΈΞ“(π‘’βˆ’π‘Ž)Ξ“(π‘’βˆ’π‘)Ξ“(2π‘βˆ’π‘’βˆ’π‘Ž+2)Ξ“((1/2)π‘’βˆ’(1/2)π‘Ž+(1/2))Ξ“(π‘βˆ’(1/2)π‘’βˆ’(1/2)π‘Ž+1)Ξ“βˆ’Ξ“((1/2)𝑒+(1/2)π‘Žβˆ’(1/2))Ξ“(π‘βˆ’(1/2)𝑒+(1/2)π‘Ž+1)((1/2)π‘’βˆ’(1/2)π‘Ž)Ξ“(π‘βˆ’(1/2)π‘’βˆ’(1/2)π‘Ž+(3/2))ξ‚ΉΞ“((1/2)𝑒+(1/2)π‘Ž)Ξ“(π‘βˆ’(1/2)𝑒+(1/2)π‘Ž+(1/2))(4.10) provided that β„œ(𝑐)>0.3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,3βˆ’π‘Ž,𝑐;1𝑒,2𝑐+2βˆ’π‘’Ξ“(𝑒)Ξ“(2π‘βˆ’π‘’+2)Ξ“(π‘’βˆ’π‘βˆ’1)22π‘Žβˆ’2(Γ—ξ‚Έπ‘βˆ’1)(π‘Žβˆ’1)(π‘Žβˆ’2)Ξ“(π‘’βˆ’π‘Ž)Ξ“(π‘’βˆ’π‘)Ξ“(2π‘βˆ’π‘’βˆ’π‘Ž+2)(2π‘βˆ’π‘’)Ξ“((1/2)π‘’βˆ’(1/2)π‘Ž+(1/2))Ξ“(π‘βˆ’(1/2)π‘’βˆ’(1/2)π‘Ž+1)Ξ“βˆ’((1/2)𝑒+(1/2)π‘Žβˆ’(3/2))Ξ“(π‘βˆ’(1/2)𝑒+(1/2)π‘Ž)(π‘’βˆ’2)Ξ“((1/2)π‘’βˆ’(1/2)π‘Ž)Ξ“(π‘βˆ’(1/2)π‘’βˆ’(1/2)π‘Ž+(3/2))ξ‚ΉΞ“((1/2)𝑒+(1/2)π‘Žβˆ’1)Ξ“(π‘βˆ’(1/2)𝑒+(1/2)π‘Žβˆ’(1/2))(4.11) provided that β„œ(𝑐)>0.

5. Main Summation Formulas

In this section, the following extensions of the classical summation theorems will be established. In all these theorems we have β„œ(𝑑)>0.(i)Extension of Kummer's theorem:3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,𝑏,𝑑+1;βˆ’12+π‘Žβˆ’π‘,𝑑Γ(1/2)Ξ“(2+π‘Žβˆ’π‘)2π‘Žξ‚Έ(1βˆ’π‘)((1+π‘Žβˆ’π‘)/𝑑)βˆ’1Ξ“+((1/2)π‘Ž)Ξ“((1/2)π‘Žβˆ’π‘+(3/2))(1βˆ’(π‘Ž/𝑑))Ξ“ξ‚Ή.((1/2)π‘Ž+(1/2))Ξ“((1/2)π‘Žβˆ’π‘+1)(5.1)(ii)Extension of Gauss's second theorem:3𝐹2⎑⎒⎒⎒⎒⎣;1π‘Ž,𝑏,𝑑+1212⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=(π‘Ž+𝑏+3),𝑑Γ(1/2)Ξ“((1/2)π‘Ž+(1/2)𝑏+(3/2))Ξ“((1/2)π‘Žβˆ’(1/2)π‘βˆ’(1/2))Γ—ξ‚»[]Ξ“((1/2)π‘Žβˆ’(1/2)𝑏+(3/2))(1/2)(π‘Ž+𝑏+1)βˆ’(π‘Žπ‘/𝑑)+[]Ξ“((1/2)π‘Ž+(1/2))Ξ“((1/2)𝑏+(1/2))((π‘Ž+𝑏+1)/𝑑)βˆ’2ξ‚Ό.Ξ“((1/2)π‘Ž)Ξ“((1/2)𝑏)(5.2)(iii)Extension of Bailey's theorem:3𝐹2⎑⎒⎒⎒⎣;1π‘Ž,1βˆ’π‘Ž,𝑑+12⎀βŽ₯βŽ₯βŽ₯⎦=𝑐+1,𝑑Γ(1/2)Ξ“(𝑐+1)2𝑐(2/𝑑)+Ξ“((1/2)𝑐+(1/2)π‘Ž)Ξ“((1/2)π‘βˆ’(1/2)π‘Ž+(1/2))(1βˆ’(𝑐/𝑑))ξ‚Ό.Ξ“((1/2)π‘βˆ’(1/2)π‘Ž+1)Ξ“((1/2)𝑐+(1/2)π‘Ž+(1/2))(5.3)(iv)Extension of Watson's theorem:

First Extension:4𝐹3⎑⎒⎒⎒⎣1π‘Ž,𝑏,𝑐,𝑑+1;12⎀βŽ₯βŽ₯βŽ₯⎦=2(π‘Ž+𝑏+1),2𝑐+1,π‘‘π‘Ž+π‘βˆ’2Ξ“(𝑐+(1/2))Ξ“((1/2)π‘Ž+(1/2)𝑏+(1/2))Ξ“(π‘βˆ’(1/2)π‘Žβˆ’(1/2)𝑏+(1/2))Γ—ξ‚»Ξ“(1/2)Ξ“(π‘Ž)Ξ“(𝑏)Ξ“((1/2)π‘Ž)Ξ“((1/2)𝑏)+Ξ“(π‘βˆ’(1/2)π‘Ž+(1/2))Ξ“(π‘βˆ’(1/2)𝑏+(1/2))((2π‘βˆ’π‘‘)/𝑑)Ξ“((1/2)π‘Ž+(1/2))Ξ“((1/2)𝑏+(1/2))Ξ“ξ‚Ό((1/2)π‘βˆ’(1/2)π‘Ž+1)Ξ“(π‘βˆ’(1/2)𝑏+1)(5.4) provided β„œ(2π‘βˆ’π‘Žβˆ’π‘)>βˆ’1.

Second Extension:4𝐹3⎑⎒⎒⎒⎣1π‘Ž,𝑏,𝑐,𝑑+1;12⎀βŽ₯βŽ₯βŽ₯⎦=2(π‘Ž+𝑏+3),2𝑐,π‘‘π‘Ž+π‘βˆ’2Ξ“(𝑐+(1/2))Ξ“((1/2)π‘Ž+(1/2)𝑏+(3/2))Ξ“(π‘βˆ’(1/2)π‘Žβˆ’(1/2)π‘βˆ’(1/2))(Γ—ξ‚»π›Όπ‘Žβˆ’π‘βˆ’1)(π‘Žβˆ’π‘+1)Ξ“(1/2)Ξ“(π‘Ž)Ξ“(𝑏)Ξ“((1/2)π‘Ž)Ξ“((1/2)𝑏)Ξ“(π‘βˆ’(1/2)π‘Ž+(1/2))Ξ“(π‘βˆ’(1/2)𝑏+(1/2))+𝛽Γ((1/2)π‘Ž+(1/2))Ξ“((1/2)𝑏+(1/2))ξ‚ΌΞ“(π‘βˆ’(1/2)π‘Ž)Ξ“(π‘βˆ’(1/2)𝑏)(5.5) provided β„œ(2π‘βˆ’π‘Žβˆ’π‘)>1, 𝛼 and 𝛽 are given by 𝛼=π‘Ž(2π‘βˆ’π‘Ž)+𝑏(2π‘βˆ’π‘)βˆ’2𝑐+1βˆ’π‘Žπ‘π‘‘ξ‚ƒ1(4π‘βˆ’π‘Žβˆ’π‘βˆ’1),𝛽=8ξ‚„.2𝑑(π‘Ž+𝑏+1)βˆ’1(5.6)(v)Extension of Dixon's theorem:4𝐹3⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π›Όπ‘Ž,𝑏,𝑐,𝑑+1;12+π‘Žβˆ’π‘,1+π‘Žβˆ’π‘,𝑑(π‘βˆ’1)Ξ“(1+π‘Žβˆ’π‘)Ξ“(2+π‘Žβˆ’π‘)Ξ“((3/2)+(1/2)π‘Žβˆ’π‘βˆ’π‘)Ξ“(1/2)2π‘Ž+𝛽Γ((1/2)π‘Ž)Ξ“((1/2)π‘Žβˆ’π‘+(1/2))Ξ“(2+π‘Žβˆ’π‘βˆ’π‘)Ξ“((1/2)π‘Žβˆ’π‘+(3/2))2(π‘βˆ’1)βˆ’π‘Žβˆ’1Ξ“(1/2)Ξ“(1+π‘Žβˆ’π‘)Ξ“(1+π‘Žβˆ’π‘)Ξ“(1+(1/2)π‘Žβˆ’π‘βˆ’π‘)Ξ“((1/2)π‘Ž+(1/2))Ξ“(1+(1/2)π‘Žβˆ’π‘)Ξ“(1+(1/2)π‘Žβˆ’π‘)Ξ“(1+π‘Žβˆ’π‘βˆ’π‘)(5.7) provided β„œ(π‘Žβˆ’2π‘βˆ’2𝑐)>βˆ’2, 𝛼 and 𝛽 are given by 1𝛼=1βˆ’π‘‘(1+π‘Žβˆ’π‘),𝛽=1+π‘Žβˆ’π‘ξ‚ƒπ‘Ž1+π‘Žβˆ’π‘βˆ’π‘π‘‘ξ‚€1(1+π‘Žβˆ’π‘βˆ’2𝑐)βˆ’22.π‘Žβˆ’π‘βˆ’π‘+1(5.8)(vi)Extension of Whipple's theorem:4𝐹3⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=2π‘Ž,1βˆ’π‘Ž,𝑐,𝑑+1;1𝑒+1,2π‘βˆ’π‘’+1,π‘‘βˆ’2π‘ŽΞ“(𝑒+1)Ξ“(π‘’βˆ’π‘)Ξ“(2π‘βˆ’π‘’+1)Γ—ξ‚»ξ‚€Ξ“(π‘’βˆ’π‘Ž+1)Ξ“(π‘’βˆ’π‘+1)Ξ“(2π‘βˆ’π‘’βˆ’π‘Ž+1)1βˆ’2π‘βˆ’π‘’π‘‘ξ‚Ξ“((1/2)π‘’βˆ’(1/2)π‘Ž+1)Ξ“(π‘βˆ’(1/2)π‘’βˆ’(1/2)π‘Ž+(1/2))+𝑒Γ((1/2)𝑒+(1/2)π‘Ž)Ξ“(π‘βˆ’(1/2)𝑒+(1/2)π‘Ž+(1/2))π‘‘ξ‚βˆ’1Ξ“((1/2)π‘’βˆ’(1/2)π‘Ž+(1/2))Ξ“(π‘βˆ’(1/2)π‘’βˆ’(1/2)π‘Ž+1)ξ‚ΌΞ“((1/2)𝑒+(1/2)π‘Ž+(1/2))Ξ“(π‘βˆ’(1/2)𝑒+(1/2)π‘Ž)(5.9) provided β„œ(𝑐)>0.(vii)Extension of (2.8):3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=ξ‚€π‘Žπ‘Ž,𝑏,1+𝑑;βˆ’11+π‘Žβˆ’π‘,𝑑1βˆ’ξ‚2𝑑Γ(1+π‘Žβˆ’π‘)Ξ“(1+(1/2)π‘Ž)+ξ‚€π‘ŽΞ“(1+π‘Ž)Ξ“(1+(1/2)π‘Žβˆ’π‘)2𝑑Γ(1+π‘Žβˆ’π‘)Ξ“((1/2)π‘Ž+(1/2)).Ξ“(1+π‘Ž)Ξ“((1/2)π‘Žβˆ’π‘+(1/2))(5.10)(viii)Extension of (2.9):4𝐹3⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=ξ‚€π‘Žπ‘Ž,𝑏,𝑐,𝑑+1;11+π‘Žβˆ’π‘,1+π‘Žβˆ’π‘,𝑑1βˆ’ξ‚2𝑑Γ(1+(1/2)π‘Ž)Ξ“(1+π‘Žβˆ’π‘)Ξ“(1+π‘Žβˆ’π‘)Ξ“(1+(1/2)π‘Žβˆ’π‘βˆ’π‘)+ξ‚€π‘ŽΞ“(1+π‘Ž)Ξ“(1+π‘Žβˆ’π‘βˆ’π‘)Ξ“(1+(1/2)π‘Žβˆ’π‘)Ξ“(1+(1/2)π‘Žβˆ’π‘)2𝑑Γ((1/2)+(1/2)π‘Ž)Ξ“(1+π‘Žβˆ’π‘)Ξ“(1+π‘Žβˆ’π‘)Ξ“((1/2)+(1/2)π‘Žβˆ’π‘βˆ’π‘)Ξ“(1+π‘Ž)Ξ“(1+π‘Žβˆ’π‘βˆ’π‘)Ξ“((1/2)+(1/2)π‘Žβˆ’π‘)Ξ“((1/2)+(1/2)π‘Žβˆ’π‘)(5.11) provided β„œ(π‘Žβˆ’2π‘βˆ’2𝑐)>βˆ’1.

5.1. Derivations

In order to derive (5.1), it is just a simple exercise to prove the following relation: 3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,𝑏,𝑑+1;βˆ’12+π‘Žβˆ’π‘,𝑑2𝐹1⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’π‘Ž,𝑏;βˆ’12+π‘Žβˆ’π‘π‘Žπ‘π‘‘(2+π‘Žβˆ’π‘)2𝐹1⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦.π‘Ž+1,𝑏+1;βˆ’13+π‘Žβˆ’π‘(5.12)

Now, it is easy to see that the first and second 2𝐹1 on the right-hand side of (5.12) can be evaluated with the help of contiguous Kummer's theorems (4.1), and after a little simplification, we arrive at the desired result (5.1).

In the exactly same manner, the results (5.2) to (5.11) can be established with the help of the following relations:3𝐹2⎑⎒⎒⎒⎒⎣;1π‘Ž,𝑏,𝑑+1212⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=(π‘Ž+𝑏+3),𝑑2𝐹1⎑⎒⎒⎒⎒⎣;1π‘Ž,𝑏212⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦+(π‘Ž+𝑏+3)π‘Žπ‘π‘‘(π‘Ž+𝑏+3)2𝐹1⎑⎒⎒⎒⎒⎣;1π‘Ž+1,𝑏+1212⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,(π‘Ž+𝑏+5)3𝐹2⎑⎒⎒⎒⎣;1π‘Ž,1βˆ’π‘Ž,𝑑+12⎀βŽ₯βŽ₯βŽ₯⎦=𝑐+1,𝑑2𝐹1⎑⎒⎒⎒⎣;1π‘Ž,1βˆ’π‘Ž2⎀βŽ₯βŽ₯βŽ₯⎦+𝑐+1π‘Ž(1βˆ’π‘Ž)2𝑑(1+𝑐)2𝐹1⎑⎒⎒⎒⎣;1π‘Ž+1,2βˆ’π‘Ž2⎀βŽ₯βŽ₯βŽ₯⎦,𝑐+24𝐹3⎑⎒⎒⎒⎣1π‘Ž,𝑏,𝑐,𝑑+1;12(⎀βŽ₯βŽ₯βŽ₯⎦=π‘Ž+𝑏+1),2𝑐+1,𝑑3𝐹2⎑⎒⎒⎒⎣1π‘Ž,𝑏,𝑐;12⎀βŽ₯βŽ₯βŽ₯⎦+(π‘Ž+𝑏+1),2𝑐+12π‘Žπ‘π‘π‘‘(2𝑐+1)(π‘Ž+𝑏+1)3𝐹2⎑⎒⎒⎒⎣1π‘Ž+1,𝑏+1,𝑐+1;12⎀βŽ₯βŽ₯βŽ₯⎦,(π‘Ž+𝑏+3),2𝑐+24𝐹3⎑⎒⎒⎒⎣1π‘Ž,𝑏,𝑐,𝑑+1;12⎀βŽ₯βŽ₯βŽ₯⎦=(π‘Ž+𝑏+3),2𝑐,𝑑3𝐹2⎑⎒⎒⎒⎣1π‘Ž,𝑏,𝑐;12⎀βŽ₯βŽ₯βŽ₯⎦+(π‘Ž+𝑏+3),2π‘π‘Žπ‘π‘‘(π‘Ž+𝑏+3)3𝐹2⎑⎒⎒⎒⎣1π‘Ž+1,𝑏+1,𝑐+1;12⎀βŽ₯βŽ₯βŽ₯⎦,(π‘Ž+𝑏+5),2𝑐+14𝐹3⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,𝑏,𝑐,𝑑+1;12+π‘Žβˆ’π‘,1+π‘Žβˆ’π‘,𝑑3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦+π‘Ž,𝑏,𝑐;12+π‘Žβˆ’π‘,1+π‘Žβˆ’π‘π‘Žπ‘π‘π‘‘(2+π‘Žβˆ’π‘)(1+π‘Žβˆ’π‘)3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,π‘Ž+1,𝑏+1,𝑐+1;13+π‘Žβˆ’π‘,2+π‘Žβˆ’π‘4𝐹3⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,1βˆ’π‘Ž,𝑐,𝑑+1;1𝑒+1,2π‘βˆ’π‘’+1,𝑑3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦+π‘Ž,1βˆ’π‘Ž,𝑐;1𝑒+1,2π‘βˆ’π‘’+1π‘Žπ‘(1βˆ’π‘Ž)𝑑(𝑒+1)(2π‘βˆ’π‘’+1)3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,π‘Ž+1,2βˆ’π‘Ž,𝑐+1;1𝑒+2,2π‘βˆ’π‘’+23𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,𝑏,𝑑+1;βˆ’11+π‘Žβˆ’π‘,𝑑2𝐹1⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’π‘Ž,𝑏;βˆ’11+π‘Žβˆ’π‘π‘Žπ‘π‘‘(1+π‘Žβˆ’π‘)2𝐹1⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,π‘Ž+1,𝑏+1;βˆ’12+π‘Žβˆ’π‘4𝐹3⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=π‘Ž,𝑏,𝑐,𝑑+1;11+π‘Žβˆ’π‘,1+π‘Žβˆ’π‘,𝑑3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦+π‘Ž,𝑏,𝑐;11+π‘Žβˆ’π‘,1+π‘Žβˆ’π‘π‘Žπ‘π‘π‘‘(1+π‘Žβˆ’π‘)(1+π‘Žβˆ’π‘)3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦π‘Ž+1,𝑏+1,𝑐+1;12+π‘Žβˆ’π‘,2+π‘Žβˆ’π‘(5.13) and using the results (4.2); (2.4), (4.3); (2.5), (4.4); (4.5), (4.6); (4.8), (4.9); (4.10), (4.11); (2.2), (4.1), and (2.6), (4.7), respectively.

5.2. Special Cases
(1)In (5.1), if we take 𝑑=1+π‘Žβˆ’π‘, we get Kummer's theorem (2.2).(2)In (5.2), if we take 𝑑=(1/2)(π‘Ž+𝑏+1), we get Gauss's second theorem (2.3).(3)In (5.3), if we take 𝑑=𝑐, we get Bailey's theorem (2.4).(4)In (5.4), if we take 𝑑=2𝑐, we get Watson's theorem (2.5).(5)In (5.5), if we take 𝑑=(1/2)(π‘Ž+𝑏+1), we again get Watson's theorem (2.5).(6)In (5.7), if we take 𝑑=1+π‘Žβˆ’π‘, we get Dixon's theorem (2.6).(7)In (5.9), if we take 𝑑=𝑒, we get Whipple's theorem (2.7).(8)In (5.10), if we take 𝑑=(1/2)π‘Ž, we get (2.8).(9)In (5.11), if we take 𝑑=(1/2)π‘Ž, we get (2.9).

6. Generalizations of Summations Due to Ramanujan

In this section, the following summations, which generalize Ramanujan's summations (3.1) to (3.11), will be established.

In all the summations, we have 𝑑>0.(i)For β„œ(π‘₯)>1/2:ξ‚€1βˆ’π‘₯βˆ’1π‘₯+2𝑑+1𝑑+ξ‚΅(π‘₯βˆ’1)(π‘₯βˆ’2)ξ‚Άξ‚€(π‘₯+2)(π‘₯+3)𝑑+2π‘‘ξ‚βˆ’β‹―=(π‘₯+1)2𝑑π‘₯(2π‘₯βˆ’1){𝑑(2π‘₯βˆ’1)+(1βˆ’π‘₯)},(6.1)ξ‚€11βˆ’22𝑑+1+ξ‚€2𝑑1β‹…32β‹…42𝑑+2ξ‚βˆš3π‘‘βˆ’β‹―=ξ‚Έξ‚€12πœ‹π‘‘ξ‚4βˆ’1Ξ“2+ξ‚€1(1/4)1βˆ’ξ‚12𝑑Γ2ξ‚Ή.(3/4)(6.2)(ii)For β„œ(π‘₯)>0:ξ‚€1βˆ’π‘₯βˆ’1π‘₯+2𝑑+1𝑑+ξ‚΅(π‘₯βˆ’1)(π‘₯βˆ’2)ξ‚Άξ‚€(π‘₯+2)(π‘₯+3)𝑑+2𝑑=βˆ’β‹―Ξ“(1/2)Ξ“(2+π‘₯)ξ‚Έξ‚€2π‘₯1+π‘₯𝑑1βˆ’1+ξ‚€1Ξ“(1/2)Ξ“(1+π‘₯)1βˆ’π‘‘ξ‚1ξ‚Ή,Ξ“(π‘₯+(1/2))(6.3)11+2ξ‚€122𝑑+1+12𝑑22ξ‚€1β‹…32β‹…42𝑑+2ξ‚βˆš3π‘‘βˆ’β‹―=πœ‹ξ‚Έ1𝑑Γ2+ξ‚€1(3/4)1βˆ’π‘‘ξ‚8Ξ“2ξ‚Ή.(1/4)(6.4)(iii)For β„œ(π‘₯)>0:11βˆ’3ξ‚€π‘₯βˆ’1π‘₯+2𝑑+1𝑑+15ξ‚΅(π‘₯βˆ’1)(π‘₯βˆ’2)ξ‚Άξ‚€(π‘₯+2)(π‘₯+3)𝑑+2𝑑=ξ‚€βˆ’β‹―1+π‘₯π‘‘ξ‚βˆ’1(π‘₯+1)+πœ‹2π‘₯(2π‘₯+1)4ξ‚€12βˆ’π‘‘ξ‚Ξ“(π‘₯)Ξ“(π‘₯+2)(2π‘₯+1)Ξ“2.(π‘₯+(1/2))(6.5)(iv)For β„œ(π‘₯)>1/4:1+(π‘₯βˆ’1)2ξ‚€(π‘₯+1)(π‘₯+2)𝑑+1𝑑+(π‘₯βˆ’1)2(π‘₯βˆ’2)2(π‘₯+1)(π‘₯+2)2ξ‚€(π‘₯+3)𝑑+2𝑑=ξ‚€+β‹―1+π‘₯𝑑2βˆ’12π‘₯βˆ’1Ξ“(π‘₯+2)Ξ“(π‘₯+(3/2))βˆšβˆ’βˆšπœ‹Ξ“(2π‘₯+1)πœ‹8(π‘₯+1)Ξ“2(π‘₯)Ξ“(2π‘₯+1)Ξ“(2π‘₯)Ξ“21(π‘₯+(1/2))𝑑.(3π‘₯βˆ’1)βˆ’4π‘₯+1(6.6)(v)For β„œ(π‘₯)>1:1βˆ’3(π‘₯βˆ’1)(ξ‚€π‘₯+2)𝑑+1𝑑+5(π‘₯βˆ’1)(π‘₯βˆ’2)(ξ‚€π‘₯+2)(π‘₯+3)𝑑+2𝑑=1+β‹―4π‘₯Ξ“(π‘₯+2)Ξ“(π‘₯+3)Ξ“2ξ‚€(π‘₯+(1/2))1βˆ’1+π‘₯𝑑,11+5ξ‚€122𝑑+1+12𝑑9ξ‚€1β‹…32β‹…42𝑑+2=33𝑑+β‹―ξ‚€14πœ‹π‘‘ξ‚+πœ‹βˆ’123Ξ“4ξ‚€1(3/4)1βˆ’ξ‚,14𝑑1+5𝑑+119𝑑2+192ξ‚΅5(𝑑+2)ξ‚Άξ‚€13𝑑1β‹…3=2β‹…4+β‹―5πœ‹3/2√482Ξ“2ξ‚€5(3/4)ξ‚βˆ’54π‘‘βˆ’1√482πœ‹5/2Ξ“2ξ‚€3(3/4)βˆ’38𝑑2,ξ‚€11+23𝑑+1+ξ‚€2𝑑1β‹…32β‹…43𝑑+2=πœ‹3𝑑+β‹―Ξ“4(βˆ’33/4)2ξ‚€11βˆ’π‘‘ξ‚Ξ“2(3/4)πœ‹3.(6.7)(vi)For β„œ(π‘₯)<2/3:𝑛1+32𝑑+1𝑑1+ξ‚΅1!𝑛(𝑛+1)3ξ‚Άξ‚€3.4𝑑+2𝑑1=(2!+β‹―1βˆ’(1/𝑑))(π‘›βˆ’1)Ξ“(1/2)Ξ“((3/2)βˆ’(2𝑛/2))2𝑛+2Ξ“(𝑛/2)Ξ“((1/2)βˆ’(𝑛/2))Ξ“((3/2)βˆ’(𝑛/2))Ξ“(2βˆ’π‘›)βˆ’π‘›βˆ’1(π‘›βˆ’1)2[](𝑛/𝑑)(1βˆ’2𝑛)βˆ’(2βˆ’3𝑛)Ξ“(1/2)Ξ“(1βˆ’(3𝑛/2))Ξ“2.(1βˆ’(𝑛/2))Ξ“((𝑛/2)+(1/2))Ξ“(1βˆ’π‘›)(6.8)(vii)For β„œ(π‘₯)>1/2:1βˆ’3(π‘₯βˆ’1)ξ‚€(π‘₯+2)𝑑+1𝑑+5(π‘₯βˆ’1)(π‘₯βˆ’2)ξ‚€(π‘₯+2)(π‘₯+3)𝑑+2𝑑π‘₯βˆ’β‹―=+ξ‚€12𝑑1βˆ’ξ‚βˆš2𝑑Γ(1+π‘₯)πœ‹.2Ξ“(π‘₯+(1/2))(6.9)(viii)For β„œ(π‘₯)>1/2:1+(π‘₯βˆ’1)2(π‘₯+1)2𝑑+1𝑑+(π‘₯βˆ’1)2(π‘₯βˆ’2)2(π‘₯+1)2(π‘₯+2)2𝑑+2𝑑=ξ‚€1+β‹―1βˆ’ξ‚βˆš2π‘‘πœ‹Ξ“2(1+π‘₯)Ξ“(2π‘₯βˆ’(1/2))2Ξ“(2π‘₯)Ξ“2+1(π‘₯+(1/2))Ξ“2𝑑2(1+π‘₯)Ξ“(2π‘₯βˆ’1)Ξ“2.(π‘₯)Ξ“(2π‘₯)(6.10)

6.1. Derivations

The series (6.1) corresponds to3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦1,1βˆ’π‘₯,1+𝑑;12+π‘₯,𝑑(6.11) which is a special case of extended Gauss's summation theorem (3.26) for π‘Ž=1,𝑏=1βˆ’π‘₯ and 𝑐=1+π‘₯.

The series (6.2) corresponds to3𝐹2⎑⎒⎒⎒⎣12,12⎀βŽ₯βŽ₯βŽ₯⎦,𝑑+1;βˆ’12,𝑑(6.12) which is a special case of extended Kummer's summation theorem (5.1) for π‘Ž=𝑏=1/2. Similarly the series (6.3) corresponds to3𝐹2⎑⎒⎒⎒⎣12⎀βŽ₯βŽ₯βŽ₯⎦,1βˆ’π‘₯,𝑑+1;βˆ’12+π‘₯,𝑑(6.13) which is a special case of extended Kummer's theorem (5.1) for π‘Ž=1 and 𝑏=1βˆ’π‘₯.

The series (6.4) corresponds to 3𝐹2⎑⎒⎒⎒⎣12,12;1,𝑑+12⎀βŽ₯βŽ₯βŽ₯⎦2,𝑑(6.14) which is a special case of extended Gauss's second summation theorem (5.2) for π‘Ž=𝑏=1/2 or extended Bailey's summation theorem (5.3) for π‘Ž=1/2, 𝑐=1.

Also, it can be easily seen that the series (6.5) to (6.8) which correspond to 4𝐹3⎑⎒⎒⎒⎒⎣11,23,1βˆ’π‘₯,1+𝑑;12⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,,2+π‘₯,𝑑4𝐹3⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,1,1βˆ’π‘₯,1+π‘₯,1+𝑑;11+π‘₯,2+π‘₯,𝑑4𝐹3⎑⎒⎒⎒⎒⎣31,21,1βˆ’π‘₯,1+𝑑;12⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,,1+π‘₯,𝑑4𝐹3⎑⎒⎒⎒⎒⎣12,12,145,1+𝑑;12,4⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,,𝑑4𝐹3⎑⎒⎒⎒⎒⎣12,14,145,1+𝑑;14,54⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,,𝑑4𝐹3⎑⎒⎒⎒⎣12,12,12⎀βŽ₯βŽ₯βŽ₯⎦,,1+𝑑;12,1,𝑑4𝐹3⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦π‘›,𝑛,𝑛,1+𝑑;11,2,𝑑(6.15) are special cases of extended Dixon's theorem (5.7).

The series (6.9) corresponds to3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦1,1βˆ’π‘₯,𝑑+1;βˆ’11+π‘₯,𝑑(6.16) which is a special case of (5.10) for π‘Ž=1 and 𝑏=1βˆ’π‘₯. And the series (6.10) corresponds to4𝐹3⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦1,1βˆ’π‘₯,1βˆ’π‘₯,1+𝑑;11+π‘₯,1+π‘₯,𝑑(6.17) which is a special case of (5.11) for π‘Ž=1, 𝑏=1βˆ’π‘₯=𝑐.

7. Concluding Remarks

(1)Various other applications of these results are under investigations and will be published later.(2)Further generalizations of the extended summation theorem (5.1) to (5.9) in the forms

3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,π‘Ž,𝑏,𝑑+1;βˆ’12+π‘Žβˆ’π‘+𝑖,𝑑3𝐹2⎑⎒⎒⎒⎒⎣;1π‘Ž,𝑏,𝑑+1212⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,(π‘Ž+𝑏+3+𝑖),𝑑3𝐹2⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦π‘Ž,1βˆ’π‘Ž+𝑖,𝑑+1;1𝑐+1,𝑑(7.1) each for 𝑖=0,Β±1,Β±2,Β±3,Β±4,Β±5, and4𝐹3⎑⎒⎒⎒⎣1π‘Ž,𝑏,𝑐,𝑑+1;12⎀βŽ₯βŽ₯βŽ₯⎦,(π‘Ž+𝑏+𝑖+1),2𝑐+𝑗,𝑑4𝐹3⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,π‘Ž,𝑏,𝑐,𝑑+1;12+π‘Žβˆ’π‘+𝑖,1+π‘Žβˆ’π‘+𝑖+𝑗,𝑑(7.2) each for 𝑖,𝑗=0,Β±1,Β±2,Β±3, and4𝐹3⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,π‘Ž,𝑏,𝑐,1+𝑑;1𝑒,𝑓,𝑑(7.3) where π‘Ž+𝑏=1+𝑖+𝑗, 𝑒+𝑓=2𝑐+𝑗 for 𝑖,𝑗=0,Β±1,Β±2,Β±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.

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