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 series𝑛=0(𝑎)𝑛(𝑏)𝑛(𝑐)𝑛𝑧𝑛𝑛!=1+𝑎𝑏1𝑐𝑧+𝑎(𝑎+1)𝑏(𝑏+1)𝑧12𝑐(𝑐+1)2+,(1.1) where (𝜆)𝑛 denotes the Pochhammer symbol defined (for 𝜆) by(𝜆)𝑛=1(𝑛=0)𝜆(𝜆+1)(𝜆+𝑛1)(𝑛={1,2,3,}).(1.2) 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 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 𝑎=𝑐 and 𝑏=1 also when 𝑏=𝑐 and 𝑎=1. The series (1.1) is represented by the notation 2𝐹1[𝑎,𝑏;𝑐;𝑧] or2𝐹1𝑐,𝑎,𝑏;𝑧(1.3) which is usually referred to as Gauss hypergeometric function. In (1.1), the three elements 𝑎, 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 𝑎 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 𝑐0,1,2,. Also when |𝑧|=1, the series is absolutely convergent if (𝑐𝑎𝑏)>0, conditionally convergent if 1<(𝑐𝑎𝑏)0, 𝑧1 and divergent if (𝑐𝑎𝑏)1.

Further, if in (1.1), we replace 𝑧 by 𝑧/𝑏 and let 𝑏, then ((𝑏)𝑛𝑧𝑛/𝑏𝑛)𝑧𝑛, and we arrive to the following Kummer's series𝑛=0(𝑎)𝑛(𝑐)𝑛𝑧𝑛𝑎𝑛!=1+1𝑐𝑧+𝑎(𝑎+1)𝑧12𝑐(𝑐+1)2+.(1.4)

This series is absolutely convergent for all values of 𝑎, 𝑐, and 𝑧, real or complex, excluding 𝑐=0,1,2, and is represented by the notation 1𝐹1(𝑎;𝑐;𝑧) or1𝐹1𝑐,𝑎,;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 𝑝𝐹𝑞 defined by𝑝𝐹𝑞𝑎1𝑎𝑝𝑏;𝑧1𝑏𝑞=𝑛=0𝑎1𝑛𝑎𝑝𝑛𝑏1𝑛𝑏𝑞𝑛𝑧𝑛.𝑛!(1.6)

The series (1.6) is convergent for all |𝑧|< if 𝑝𝑞 and for |𝑧|<1 if 𝑝=𝑞+1 while it is divergent for all 𝑧, 𝑧0 if 𝑝>𝑞+1. When |𝑧|=1 with 𝑝=𝑞+1, the series (1.6) converges absolutely if𝑞𝑗=1𝑏𝑗𝑝𝑗=1𝑎𝑗>0,(1.7) conditionally convergent if1<𝑞𝑗=1𝑏𝑗𝑝𝑗=1𝑎𝑗0,𝑧1(1.8) and divergent if𝑞𝑗=1𝑏𝑗𝑝𝑗=1𝑎𝑗1.(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]:2𝐹1𝑐=𝑎,𝑏;1Γ(𝑐)Γ(𝑐𝑎𝑏)Γ(𝑐𝑎)Γ(𝑐𝑏)(2.1) provided (𝑐𝑎𝑏)>0.

Kummer theorem [5]:2𝐹1=𝑎,𝑏;11+𝑎𝑏Γ(1+𝑎𝑏)Γ(1+(1/2)𝑎).Γ(1+(1/2)𝑎𝑏)Γ(1+𝑎)(2.2)

Gauss’s second theorem [5]:2𝐹1;1𝑎,𝑏212=(𝑎+𝑏+1)Γ(1/2)Γ((1/2)𝑎+(1/2)𝑏+(1/2)).Γ((1/2)𝑎+(1/2))Γ((1/2)𝑏+(1/2))(2.3)

Bailey theorem [5]:2𝐹1;1𝑎,1𝑎2𝑐=Γ((1/2)𝑐)Γ((1/2)𝑐+(1/2)).Γ((1/2)𝑐+(1/2)𝑎)Γ((1/2)𝑐(1/2)𝑎+(1/2))(2.4)

Watson theorem [5]:3𝐹21𝑎,𝑏,𝑐;12=(𝑎+𝑏+1),2𝑐Γ(1/2)Γ(𝑐+(1/2))Γ((1/2)𝑎+(1/2)𝑏+(1/2))Γ(𝑐(1/2)𝑎(1/2)𝑏+(1/2))Γ((1/2)𝑎+(1/2))Γ((1/2)𝑏+(1/2))Γ(𝑐(1/2)𝑎+(1/2))Γ(𝑐(1/2)𝑏+(1/2))(2.5) provided (2𝑐𝑎𝑏)>1.

Dixon theorem [5]:3𝐹2=𝑎,𝑏,𝑐;11+𝑎𝑏,1+𝑎𝑐Γ(1+(1/2)𝑎)Γ(1+𝑎𝑏)Γ(1+𝑎𝑐)Γ(1+(1/2)𝑎𝑏𝑐)Γ(1+𝑎)Γ(1+(1/2)𝑎𝑏)Γ(1+(1/2)𝑎𝑐)Γ(1+𝑎𝑏𝑐)(2.6) provided (𝑎2𝑏2𝑐)>2.

Whipple theorem [5]:3𝐹2=𝑎,𝑏,𝑐;1𝑒,𝑓𝜋Γ(𝑒)Γ(𝑓)22𝑐1Γ((1/2)𝑎+(1/2)𝑒)Γ((1/2)𝑎+(1/2)𝑓)Γ((1/2)𝑏+(1/2)𝑒)Γ((1/2)𝑏+(1/2)𝑓)(2.7) provided (𝑐)>0 and (𝑒+𝑓𝑎𝑏𝑐)>0 with 𝑎+𝑏=1 and 𝑒+𝑓=2𝑐+1.

Other hypergeometric identities [5]:3𝐹21𝑎,1+21𝑎,𝑏;12=𝑎,1+𝑎𝑏Γ(1+𝑎𝑏)Γ((1/2)𝑎+(1/2)),Γ(1+𝑎)Γ((1/2)𝑎𝑏+(1/2))(2.8)4𝐹31𝑎,1+21𝑎,𝑏,𝑐;12=Γ𝑎,1+𝑎𝑏,1+𝑎𝑐(1+𝑎𝑏)Γ(1+𝑎𝑐)Γ((1/2)𝑎+(1/2))Γ((1/2)𝑎𝑏𝑐+(1/2))Γ(1+𝑎)Γ(1+𝑎𝑏𝑐)Γ((1/2)𝑎𝑏+(1/2))Γ((1/2)𝑎𝑐+(1/2))(2.9) provided (𝑎2𝑏2𝑐)>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, Saalschütz, 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 (𝑥)>1/2,1(𝑥1)(+𝑥+1)(𝑥1)(𝑥2)(𝑥𝑥+1)(𝑥+2)=,2𝑥1(3.1)1122+132421352462+=𝜋2Γ2.(3/4)(3.2)(ii)For (𝑥)>0,1+(𝑥1)+(𝑥+1)(𝑥1)(𝑥2)2(𝑥+1)(𝑥+2)+=2𝑥1Γ2(𝑥+1),Γ(2𝑥+1)(3.3)11+2122+12213242+1231352462+=𝜋Γ2.(3/4)(3.4)(iii)For (𝑥)>0,113(𝑥1)+1(𝑥+1)5(𝑥1)(𝑥2)2(𝑥+1)(𝑥+2)=4𝑥Γ4(𝑥+1)4𝑥Γ2(.2𝑥+1)(3.5)(iv)For (𝑥)>1/4,1+(𝑥1)2(𝑥+1)2+(𝑥1)(𝑥2)(𝑥+1)(𝑥+2)2+=2𝑥Γ4𝑥14(𝑥+1)Γ(4𝑥+1)Γ4.(2𝑥+1)(3.6)(v)For (𝑥)>1,13(𝑥1)(𝑥+1)+5(𝑥1)(𝑥2)(𝑥+1)(𝑥+2)=0,(3.7)11+5122+1913242𝜋+=24Γ4,1(3/4)1+5212+19213𝜋24+=5/282Γ2,1(3/4)1+22+13242𝜋+=Γ4.(3/4)(3.8)(vi)For (𝑥)<2/3,𝑥1+1!3+𝑥(𝑥+1)2!3+=6sin(𝜋𝑥/2)sin(𝜋𝑥)Γ3((1/2)𝑥+1)𝜋2𝑥2Γ.((3/2)𝑥+1)(1+2cos𝜋𝑥)(3.9)(vii)For (𝑥)>1/2,1+3(𝑥1)(𝑥+1)+5(𝑥1)(𝑥2)(𝑥+1)(𝑥+2)+=𝑥.(3.10)(viii)For (𝑥)>1/2,1+3(𝑥1)(𝑥+1)2+5(𝑥1)(𝑥2)(𝑥+1)(𝑥+2)2𝑥+=2.2𝑥1(3.11) We now come to the derivations of these summation in brief.

It is easy to see that the series (3.1) corresponds to2𝐹11,1𝑥;11+𝑥(3.12) which is a special case of Gauss's summation theorem (2.1) for 𝑎=1, 𝑏=1𝑥 and 𝑐=1+𝑥.

The series (3.2) corresponds to2𝐹112,121;1(3.13) which is a special case of Kummer's summation theorem (2.2) for 𝑎=𝑏=1/2. Similarly the series (3.3) corresponds to2𝐹11,1𝑥;11+𝑥(3.14) which is a special case of Kummer's summation theorem (2.2) for 𝑎=1, 𝑏=1𝑥.

The series (3.4) corresponds to2𝐹112,12;121(3.15) which is a special case of Gauss's second summation theorem (2.3) for 𝑎=𝑏=1/2 or Bailey's summation theorem (2.4) for 𝑎=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: 3𝐹211,23,1𝑥;12,,1+𝑥3𝐹2,1,1𝑥,1𝑥;11+𝑥,1+𝑥3𝐹231,21,1𝑥;12,,1+𝑥3𝐹212,12,145;11,4,3𝐹212,14,145;14,54,3𝐹212,12,12,;11,13𝐹2,𝑥,𝑥,𝑥;11,1(3.16) which are special cases of classical Dixon's theorem (2.6) for (i) 𝑎=1, 𝑏=1/2, 𝑐=1𝑥, (ii) 𝑎=1, 𝑏=𝑐=1𝑥, (iii) 𝑎=1, 𝑏=3/2, 𝑐=1𝑥, (iv) 𝑎=𝑏=1/2, 𝑐=1/4, (v) 𝑎=1/2, 𝑏=𝑐=1/4, (vi) 𝑎=𝑏=𝑐=1/2, and (vii) 𝑎=𝑏=𝑐=𝑥, respectively.

The series (3.10) which corresponds to3𝐹231,21,1𝑥;12,1+𝑥(3.17) is a special case of (2.8) for 𝑎=1, 𝑏=1𝑥, and the series (3.11) which corresponds to4𝐹331,21,1𝑥,1𝑥;12,1+𝑥,1+𝑥(3.18) is a special case of (2.9) for 𝑎=1, 𝑏=𝑐=1𝑥.

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. [79], 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,𝑎,𝑏;11+𝑎𝑏+𝑖(3.19)2𝐹1;1𝑎,𝑏212,(𝑎+𝑏+𝑖+1)(3.20)2𝐹1;1𝑎,1𝑎+𝑖2𝑐(3.21) each for 𝑖=0,±1,±2,±3,±4,±5, and3𝐹21𝑎,𝑏,𝑐;12(𝑎+𝑏+𝑖+1),2𝑐+𝑗(3.22) for 𝑖,𝑗=0,±1,±23𝐹2𝑎,𝑏,𝑐;11+𝑎𝑏+𝑖,1+𝑎𝑐+𝑖+𝑗(3.23) for 𝑖=3,2,1,0,1,2; 𝑗=0,1,2,3, and3𝐹2𝑎,𝑏,𝑐;1𝑒,𝑓(3.24) for 𝑎+𝑏=1+𝑖+𝑗, 𝑒+𝑓=2𝑐+1+𝑖 for 𝑖,𝑗=0,±1,±2,±3.

Notice that, if we denote (3.23) by 𝑓𝑖,𝑗, the natural symmetry𝑓𝑖,𝑗(𝑎,𝑏,𝑐)=𝑓𝑖+𝑗,𝑗(𝑎,𝑐,𝑏)(3.25) makes it possible to extend the result to 𝑗=1,2,3.

It is very interesting to mention here that, in order to complete the results (3.23) of 7×7 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=𝑎,𝑏,𝑑+1;1𝑐+1,𝑑Γ(𝑐+1)Γ(𝑐𝑎𝑏)Γ(𝑐𝑎+1)Γ(𝑐𝑏+1)(𝑐𝑎𝑏)+𝑎𝑏𝑑(3.26) provided (𝑐𝑎𝑏)>0 and (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 Kampé de Fériet 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. [79].

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. [79] will be required in our present investigations.(i)Contiguous Kummer's theorem [9]:2𝐹1=𝑎,𝑏;12+𝑎𝑏Γ(1/2)Γ(2+𝑎𝑏)2𝑎×1(1𝑏)1Γ((1/2)𝑎+(1/2))Γ((1/2)𝑎𝑏+1),Γ((1/2)𝑎)Γ((1/2)𝑎𝑏+(3/2))2𝐹1=𝑎,𝑏;13+𝑎𝑏Γ(1/2)Γ(3+𝑎𝑏)2𝑎×(1𝑏)(2𝑏)(1+𝑎𝑏)Γ2((1/2)𝑎+(1/2))Γ((1/2)𝑎𝑏+2)Γ.((1/2)𝑎)Γ((1/2)𝑎𝑏+(3/2))(4.1)(ii)Contiguous Gauss's Second theorem [9]:2𝐹1;1𝑎,𝑏212=(𝑎+𝑏+3)Γ(1/2)Γ((1/2)𝑎+(1/2)𝑏+(3/2))Γ((1/2)𝑎(1/2)𝑏(1/2))×Γ((1/2)𝑎(1/2)𝑏+(3/2))(1/2)(𝑎+𝑏1)2Γ((1/2)𝑎+(1/2))Γ((1/2)𝑏+(1/2)).Γ((1/2)𝑎)Γ((1/2)𝑎)(4.2)(iii)Contiguous Bailey's theorem [9]:2𝐹1;1𝑎,3𝑎2𝑐=Γ(1/2)Γ(𝑐)Γ(1𝑎)2𝑐3×Γ(3𝑎)(𝑐2)2Γ((1/2)𝑐(1/2)𝑎+(1/2))Γ((1/2)𝑐+(1/2)𝑎1).Γ((1/2)𝑐(1/2)𝑎)Γ((1/2)𝑐+(1/2)𝑎(3/2))(4.3)(iv)Contiguous Watson's theorem [7]: 3𝐹21𝑎,𝑏,𝑐;12=2(𝑎+𝑏+1),2𝑐+1𝑎+𝑏2Γ(𝑐+(1/2))Γ((1/2)𝑎+(1/2)𝑏+(1/2))Γ(𝑐(1/2)𝑎(1/2)𝑏+(1/2))×Γ(1/2)Γ(𝑎)Γ(𝑏)Γ((1/2)𝑎)Γ((1/2)𝑏)Γ(𝑐(1/2)𝑎+(1/2))Γ(𝑐(1/2)𝑏+(1/2))Γ((1/2)𝑎+(1/2))Γ((1/2)𝑏+(1/2))Γ(𝑐(1/2)𝑎+1)Γ(𝑐(1/2)𝑏+1)(4.4) provided that (2𝑐𝑎𝑏)>1.3𝐹21𝑎,𝑏,𝑐;12=2(𝑎+𝑏+3),2𝑐𝑎+𝑏+1Γ(𝑐+(1/2))Γ((1/2)𝑎+(1/2)𝑏+(3/2))Γ(𝑐(1/2)𝑎(1/2)𝑏(1/2))(×𝑎𝑏1)(𝑎𝑏+1)Γ(1/2)Γ(𝑎)Γ(𝑏)𝑎(2𝑐𝑎)+𝑏(2𝑐𝑏)2𝑐+18Γ((1/2)𝑎)Γ((1/2)𝑏)Γ(𝑐(1/2)𝑎+(1/2))Γ(𝑐(1/2)𝑏+(1/2))Γ((1/2)𝑎+(1/2))Γ((1/2)𝑏+(1/2))Γ(𝑐(1/2)𝑎)Γ(𝑐(1/2)𝑏)(4.5) provided that (2𝑐𝑎𝑏)>1.3𝐹21𝑎,𝑏,𝑐;12=2(𝑎+𝑏+3),2𝑐1𝑎+𝑏1Γ(𝑐(1/2))Γ((1/2)𝑎+(1/2)𝑏+(3/2))Γ(𝑐(1/2)𝑎(1/2)𝑏(1/2))(×𝑎𝑏1)(𝑎𝑏+1)Γ(1/2)Γ(𝑎)Γ(𝑏)(𝑎+𝑏1)Γ((1/2)𝑎)Γ((1/2)𝑏)Γ(𝑐(1/2)𝑎(1/2))Γ(𝑐(1/2)𝑏(1/2))(4𝑐𝑎𝑏3)Γ((1/2)𝑎+(1/2))Γ((1/2)𝑏+(1/2))Γ(𝑐(1/2)𝑎)Γ(𝑐(1/2)𝑏)(4.6) provided that (2𝑐𝑎𝑏)>1.(v)Contiguous Dixon's theorem [8]:3𝐹2=2𝑎,𝑏,𝑐;12+𝑎𝑏,2+𝑎𝑐2𝑐+1Γ(2+𝑎𝑏)Γ(2+𝑎𝑐)×(𝑏1)(𝑐1)Γ(𝑎2𝑐+2)Γ(𝑎𝑏𝑐+2)Γ((1/2)𝑎𝑐+(3/2))Γ((1/2)𝑎𝑏𝑐+2)ΓΓ((1/2)𝑎+(1/2))Γ((1/2)𝑎𝑏+1)((1/2)𝑎𝑐+1)Γ((1/2)𝑎𝑏𝑐+(5/2))Γ((1/2)𝑎)Γ((1/2)𝑎𝑏+(3/2))(4.7) provided that (𝑎2𝑏2𝑐)>4.3𝐹2=2𝑎,𝑏,𝑐;12+𝑎𝑏,1+𝑎𝑐2𝑏+1Γ(1+𝑎𝑐)Γ(2+𝑎𝑏)×(𝑏1)Γ(𝑎2𝑏+2)Γ(𝑎𝑏𝑐+2)Γ((1/2)𝑎𝑏+1)Γ((1/2)𝑎𝑏𝑐+(3/2))ΓΓ((1/2)𝑎)Γ((1/2)𝑎𝑐+(1/2))((1/2)𝑎𝑏+(3/2))Γ((1/2)𝑎𝑏𝑐+2)Γ((1/2)𝑎+(1/2))Γ((1/2)𝑎𝑐+1)(4.8) provided that (𝑎2𝑏2𝑐)>3.3𝐹2=2𝑎,𝑏,𝑐;12+𝑎𝑐,3+𝑎𝑏2𝑏+2Γ(2+𝑎𝑐)Γ(3+𝑎𝑏)×(𝑏1)(𝑏2)(𝑐1)Γ(𝑎2𝑏+3)Γ(𝑎𝑏𝑐+3)(𝑎2𝑐𝑏+3)Γ((1/2)𝑎𝑏+2)Γ((1/2)𝑎𝑏𝑐+(5/2))Γ((1/2)𝑎)Γ((1/2)𝑎𝑐+(3/2))(𝑎𝑏+1)Γ((1/2)𝑎𝑏+(3/2))Γ((1/2)𝑎𝑏𝑐+3)Γ((1/2)𝑎+(1/2))Γ((1/2)𝑎𝑐+1)(4.9) provided that (𝑎2𝑏2𝑐)>3.(vi)Contiguous Whipple's theorem [9]:3𝐹2=𝑎,1𝑎,𝑐;1𝑒,2𝑐+2𝑒Γ(𝑒)Γ(2𝑐+2𝑒)Γ(𝑒𝑐1)22𝑎×Γ(𝑒𝑎)Γ(𝑒𝑐)Γ(2𝑐𝑒𝑎+2)Γ((1/2)𝑒(1/2)𝑎+(1/2))Γ(𝑐(1/2)𝑒(1/2)𝑎+1)ΓΓ((1/2)𝑒+(1/2)𝑎(1/2))Γ(𝑐(1/2)𝑒+(1/2)𝑎+1)((1/2)𝑒(1/2)𝑎)Γ(𝑐(1/2)𝑒(1/2)𝑎+(3/2))Γ((1/2)𝑒+(1/2)𝑎)Γ(𝑐(1/2)𝑒+(1/2)𝑎+(1/2))(4.10) provided that (𝑐)>0.3𝐹2=𝑎,3𝑎,𝑐;1𝑒,2𝑐+2𝑒Γ(𝑒)Γ(2𝑐𝑒+2)Γ(𝑒𝑐1)22𝑎2(×𝑐1)(𝑎1)(𝑎2)Γ(𝑒𝑎)Γ(𝑒𝑐)Γ(2𝑐𝑒𝑎+2)(2𝑐𝑒)Γ((1/2)𝑒(1/2)𝑎+(1/2))Γ(𝑐(1/2)𝑒(1/2)𝑎+1)Γ((1/2)𝑒+(1/2)𝑎(3/2))Γ(𝑐(1/2)𝑒+(1/2)𝑎)(𝑒2)Γ((1/2)𝑒(1/2)𝑎)Γ(𝑐(1/2)𝑒(1/2)𝑎+(3/2))Γ((1/2)𝑒+(1/2)𝑎1)Γ(𝑐(1/2)𝑒+(1/2)𝑎(1/2))(4.11) provided that (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 (𝑑)>0.(i)Extension of Kummer's theorem:3𝐹2=𝑎,𝑏,𝑑+1;12+𝑎𝑏,𝑑Γ(1/2)Γ(2+𝑎𝑏)2𝑎(1𝑏)((1+𝑎𝑏)/𝑑)1Γ+((1/2)𝑎)Γ((1/2)𝑎𝑏+(3/2))(1(𝑎/𝑑))Γ.((1/2)𝑎+(1/2))Γ((1/2)𝑎𝑏+1)(5.1)(ii)Extension of Gauss's second theorem:3𝐹2;1𝑎,𝑏,𝑑+1212=(𝑎+𝑏+3),𝑑Γ(1/2)Γ((1/2)𝑎+(1/2)𝑏+(3/2))Γ((1/2)𝑎(1/2)𝑏(1/2))×[]Γ((1/2)𝑎(1/2)𝑏+(3/2))(1/2)(𝑎+𝑏+1)(𝑎𝑏/𝑑)+[]Γ((1/2)𝑎+(1/2))Γ((1/2)𝑏+(1/2))((𝑎+𝑏+1)/𝑑)2.Γ((1/2)𝑎)Γ((1/2)𝑏)(5.2)(iii)Extension of Bailey's theorem:3𝐹2;1𝑎,1𝑎,𝑑+12=𝑐+1,𝑑Γ(1/2)Γ(𝑐+1)2𝑐(2/𝑑)+Γ((1/2)𝑐+(1/2)𝑎)Γ((1/2)𝑐(1/2)𝑎+(1/2))(1(𝑐/𝑑)).Γ((1/2)𝑐(1/2)𝑎+1)Γ((1/2)𝑐+(1/2)𝑎+(1/2))(5.3)(iv)Extension of Watson's theorem:

First Extension:4𝐹31𝑎,𝑏,𝑐,𝑑+1;12=2(𝑎+𝑏+1),2𝑐+1,𝑑𝑎+𝑏2Γ(𝑐+(1/2))Γ((1/2)𝑎+(1/2)𝑏+(1/2))Γ(𝑐(1/2)𝑎(1/2)𝑏+(1/2))×Γ(1/2)Γ(𝑎)Γ(𝑏)Γ((1/2)𝑎)Γ((1/2)𝑏)+Γ(𝑐(1/2)𝑎+(1/2))Γ(𝑐(1/2)𝑏+(1/2))((2𝑐𝑑)/𝑑)Γ((1/2)𝑎+(1/2))Γ((1/2)𝑏+(1/2))Γ((1/2)𝑐(1/2)𝑎+1)Γ(𝑐(1/2)𝑏+1)(5.4) provided (2𝑐𝑎𝑏)>1.

Second Extension:4𝐹31𝑎,𝑏,𝑐,𝑑+1;12=2(𝑎+𝑏+3),2𝑐,𝑑𝑎+𝑏2Γ(𝑐+(1/2))Γ((1/2)𝑎+(1/2)𝑏+(3/2))Γ(𝑐(1/2)𝑎(1/2)𝑏(1/2))(×𝛼𝑎𝑏1)(𝑎𝑏+1)Γ(1/2)Γ(𝑎)Γ(𝑏)Γ((1/2)𝑎)Γ((1/2)𝑏)Γ(𝑐(1/2)𝑎+(1/2))Γ(𝑐(1/2)𝑏+(1/2))+𝛽Γ((1/2)𝑎+(1/2))Γ((1/2)𝑏+(1/2))Γ(𝑐(1/2)𝑎)Γ(𝑐(1/2)𝑏)(5.5) provided (2𝑐𝑎𝑏)>1, 𝛼 and 𝛽 are given by 𝛼=𝑎(2𝑐𝑎)+𝑏(2𝑐𝑏)2𝑐+1𝑎𝑏𝑑1(4𝑐𝑎𝑏1),𝛽=8.2𝑑(𝑎+𝑏+1)1(5.6)(v)Extension of Dixon's theorem:4𝐹3=𝛼𝑎,𝑏,𝑐,𝑑+1;12+𝑎𝑏,1+𝑎𝑐,𝑑(𝑏1)Γ(1+𝑎𝑐)Γ(2+𝑎𝑏)Γ((3/2)+(1/2)𝑎𝑏𝑐)Γ(1/2)2𝑎+𝛽Γ((1/2)𝑎)Γ((1/2)𝑎𝑐+(1/2))Γ(2+𝑎𝑏𝑐)Γ((1/2)𝑎𝑏+(3/2))2(𝑏1)𝑎1Γ(1/2)Γ(1+𝑎𝑐)Γ(1+𝑎𝑏)Γ(1+(1/2)𝑎𝑏𝑐)Γ((1/2)𝑎+(1/2))Γ(1+(1/2)𝑎𝑏)Γ(1+(1/2)𝑎𝑐)Γ(1+𝑎𝑏𝑐)(5.7) provided (𝑎2𝑏2𝑐)>2, 𝛼 and 𝛽 are given by 1𝛼=1𝑑(1+𝑎𝑏),𝛽=1+𝑎𝑏𝑎1+𝑎𝑏𝑐𝑑1(1+𝑎𝑏2𝑐)22.𝑎𝑏𝑐+1(5.8)(vi)Extension of Whipple's theorem:4𝐹3=2𝑎,1𝑎,𝑐,𝑑+1;1𝑒+1,2𝑐𝑒+1,𝑑2𝑎Γ(𝑒+1)Γ(𝑒𝑐)Γ(2𝑐𝑒+1)×Γ(𝑒𝑎+1)Γ(𝑒𝑐+1)Γ(2𝑐𝑒𝑎+1)12𝑐𝑒𝑑Γ((1/2)𝑒(1/2)𝑎+1)Γ(𝑐(1/2)𝑒(1/2)𝑎+(1/2))+𝑒Γ((1/2)𝑒+(1/2)𝑎)Γ(𝑐(1/2)𝑒+(1/2)𝑎+(1/2))𝑑1Γ((1/2)𝑒(1/2)𝑎+(1/2))Γ(𝑐(1/2)𝑒(1/2)𝑎+1)Γ((1/2)𝑒+(1/2)𝑎+(1/2))Γ(𝑐(1/2)𝑒+(1/2)𝑎)(5.9) provided (𝑐)>0.(vii)Extension of (2.8):3𝐹2=𝑎𝑎,𝑏,1+𝑑;11+𝑎𝑏,𝑑12𝑑Γ(1+𝑎𝑏)Γ(1+(1/2)𝑎)+𝑎Γ(1+𝑎)Γ(1+(1/2)𝑎𝑏)2𝑑Γ(1+𝑎𝑏)Γ((1/2)𝑎+(1/2)).Γ(1+𝑎)Γ((1/2)𝑎𝑏+(1/2))(5.10)(viii)Extension of (2.9):4𝐹3=𝑎𝑎,𝑏,𝑐,𝑑+1;11+𝑎𝑏,1+𝑎𝑐,𝑑12𝑑Γ(1+(1/2)𝑎)Γ(1+𝑎𝑏)Γ(1+𝑎𝑐)Γ(1+(1/2)𝑎𝑏𝑐)+𝑎Γ(1+𝑎)Γ(1+𝑎𝑏𝑐)Γ(1+(1/2)𝑎𝑏)Γ(1+(1/2)𝑎𝑐)2𝑑Γ((1/2)+(1/2)𝑎)Γ(1+𝑎𝑏)Γ(1+𝑎𝑐)Γ((1/2)+(1/2)𝑎𝑏𝑐)Γ(1+𝑎)Γ(1+𝑎𝑏𝑐)Γ((1/2)+(1/2)𝑎𝑏)Γ((1/2)+(1/2)𝑎𝑐)(5.11) provided (𝑎2𝑏2𝑐)>1.

5.1. Derivations

In order to derive (5.1), it is just a simple exercise to prove the following relation: 3𝐹2=𝑎,𝑏,𝑑+1;12+𝑎𝑏,𝑑2𝐹1𝑎,𝑏;12+𝑎𝑏𝑎𝑏𝑑(2+𝑎𝑏)2𝐹1.𝑎+1,𝑏+1;13+𝑎𝑏(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;1𝑎,𝑏,𝑑+1212=(𝑎+𝑏+3),𝑑2𝐹1;1𝑎,𝑏212+(𝑎+𝑏+3)𝑎𝑏𝑑(𝑎+𝑏+3)2𝐹1;1𝑎+1,𝑏+1212,(𝑎+𝑏+5)3𝐹2;1𝑎,1𝑎,𝑑+12=𝑐+1,𝑑2𝐹1;1𝑎,1𝑎2+𝑐+1𝑎(1𝑎)2𝑑(1+𝑐)2𝐹1;1𝑎+1,2𝑎2,𝑐+24𝐹31𝑎,𝑏,𝑐,𝑑+1;12(=𝑎+𝑏+1),2𝑐+1,𝑑3𝐹21𝑎,𝑏,𝑐;12+(𝑎+𝑏+1),2𝑐+12𝑎𝑏𝑐𝑑(2𝑐+1)(𝑎+𝑏+1)3𝐹21𝑎+1,𝑏+1,𝑐+1;12,(𝑎+𝑏+3),2𝑐+24𝐹31𝑎,𝑏,𝑐,𝑑+1;12=(𝑎+𝑏+3),2𝑐,𝑑3𝐹21𝑎,𝑏,𝑐;12+(𝑎+𝑏+3),2𝑐𝑎𝑏𝑑(𝑎+𝑏+3)3𝐹21𝑎+1,𝑏+1,𝑐+1;12,(𝑎+𝑏+5),2𝑐+14𝐹3=𝑎,𝑏,𝑐,𝑑+1;12+𝑎𝑏,1+𝑎𝑐,𝑑3𝐹2+𝑎,𝑏,𝑐;12+𝑎𝑏,1+𝑎𝑐𝑎𝑏𝑐𝑑(2+𝑎𝑏)(1+𝑎𝑐)3𝐹2,𝑎+1,𝑏+1,𝑐+1;13+𝑎𝑏,2+𝑎𝑐4𝐹3=𝑎,1𝑎,𝑐,𝑑+1;1𝑒+1,2𝑐𝑒+1,𝑑3𝐹2+𝑎,1𝑎,𝑐;1𝑒+1,2𝑐𝑒+1𝑎𝑐(1𝑎)𝑑(𝑒+1)(2𝑐𝑒+1)3𝐹2,𝑎+1,2𝑎,𝑐+1;1𝑒+2,2𝑐𝑒+23𝐹2=𝑎,𝑏,𝑑+1;11+𝑎𝑏,𝑑2𝐹1𝑎,𝑏;11+𝑎𝑏𝑎𝑏𝑑(1+𝑎𝑏)2𝐹1,𝑎+1,𝑏+1;12+𝑎𝑏4𝐹3=𝑎,𝑏,𝑐,𝑑+1;11+𝑎𝑏,1+𝑎𝑐,𝑑3𝐹2+𝑎,𝑏,𝑐;11+𝑎𝑏,1+𝑎𝑐𝑎𝑏𝑐𝑑(1+𝑎𝑏)(1+𝑎𝑐)3𝐹2𝑎+1,𝑏+1,𝑐+1;12+𝑎𝑏,2+𝑎𝑐(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+𝑎𝑏, we get Kummer's theorem (2.2).(2)In (5.2), if we take 𝑑=(1/2)(𝑎+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)(𝑎+𝑏+1), we again get Watson's theorem (2.5).(6)In (5.7), if we take 𝑑=1+𝑎𝑏, 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)𝑎, we get (2.8).(9)In (5.11), if we take 𝑑=(1/2)𝑎, 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 (𝑥)>1/2:1𝑥1𝑥+2𝑑+1𝑑+(𝑥1)(𝑥2)(𝑥+2)(𝑥+3)𝑑+2𝑑=(𝑥+1)2𝑑𝑥(2𝑥1){𝑑(2𝑥1)+(1𝑥)},(6.1)1122𝑑+1+2𝑑13242𝑑+23𝑑=12𝜋𝑑41Γ2+1(1/4)112𝑑Γ2.(3/4)(6.2)(ii)For (𝑥)>0:1𝑥1𝑥+2𝑑+1𝑑+(𝑥1)(𝑥2)(𝑥+2)(𝑥+3)𝑑+2𝑑=Γ(1/2)Γ(2+𝑥)2𝑥1+𝑥𝑑11+1Γ(1/2)Γ(1+𝑥)1𝑑1,Γ(𝑥+(1/2))(6.3)11+2122𝑑+1+12𝑑2213242𝑑+23𝑑=𝜋1𝑑Γ2+1(3/4)1𝑑8Γ2.(1/4)(6.4)(iii)For (𝑥)>0:113𝑥1𝑥+2𝑑+1𝑑+15(𝑥1)(𝑥2)(𝑥+2)(𝑥+3)𝑑+2𝑑=1+𝑥𝑑1(𝑥+1)+𝜋2𝑥(2𝑥+1)412𝑑Γ(𝑥)Γ(𝑥+2)(2𝑥+1)Γ2.(𝑥+(1/2))(6.5)(iv)For (𝑥)>1/4:1+(𝑥1)2(𝑥+1)(𝑥+2)𝑑+1𝑑+(𝑥1)2(𝑥2)2(𝑥+1)(𝑥+2)2(𝑥+3)𝑑+2𝑑=+1+𝑥𝑑212𝑥1Γ(𝑥+2)Γ(𝑥+(3/2))𝜋Γ(2𝑥+1)𝜋8(𝑥+1)Γ2(𝑥)Γ(2𝑥+1)Γ(2𝑥)Γ21(𝑥+(1/2))𝑑.(3𝑥1)4𝑥+1(6.6)(v)For (𝑥)>1:13(𝑥1)(𝑥+2)𝑑+1𝑑+5(𝑥1)(𝑥2)(𝑥+2)(𝑥+3)𝑑+2𝑑=1+4𝑥Γ(𝑥+2)Γ(𝑥+3)Γ2(𝑥+(1/2))11+𝑥𝑑,11+5122𝑑+1+12𝑑913242𝑑+2=33𝑑+14𝜋𝑑+𝜋123Γ41(3/4)1,14𝑑1+5𝑑+119𝑑2+1925(𝑑+2)13𝑑13=24+5𝜋3/2482Γ25(3/4)54𝑑1482𝜋5/2Γ23(3/4)38𝑑2,11+23𝑑+1+2𝑑13243𝑑+2=𝜋3𝑑+Γ4(33/4)211𝑑Γ2(3/4)𝜋3.(6.7)(vi)For (𝑥)<2/3:𝑛1+32𝑑+1𝑑1+1!𝑛(𝑛+1)33.4𝑑+2𝑑1=(2!+1(1/𝑑))(𝑛1)Γ(1/2)Γ((3/2)(2𝑛/2))2𝑛+2Γ(𝑛/2)Γ((1/2)(𝑛/2))Γ((3/2)(𝑛/2))Γ(2𝑛)𝑛1(𝑛1)2[](𝑛/𝑑)(12𝑛)(23𝑛)Γ(1/2)Γ(1(3𝑛/2))Γ2.(1(𝑛/2))Γ((𝑛/2)+(1/2))Γ(1𝑛)(6.8)(vii)For (𝑥)>1/2:13(𝑥1)(𝑥+2)𝑑+1𝑑+5(𝑥1)(𝑥2)(𝑥+2)(𝑥+3)𝑑+2𝑑𝑥=+12𝑑12𝑑Γ(1+𝑥)𝜋.2Γ(𝑥+(1/2))(6.9)(viii)For (𝑥)>1/2:1+(𝑥1)2(𝑥+1)2𝑑+1𝑑+(𝑥1)2(𝑥2)2(𝑥+1)2(𝑥+2)2𝑑+2𝑑=1+12𝑑𝜋Γ2(1+𝑥)Γ(2𝑥(1/2))2Γ(2𝑥)Γ2+1(𝑥+(1/2))Γ2𝑑2(1+𝑥)Γ(2𝑥1)Γ2.(𝑥)Γ(2𝑥)(6.10)

6.1. Derivations

The series (6.1) corresponds to3𝐹21,1𝑥,1+𝑑;12+𝑥,𝑑(6.11) which is a special case of extended Gauss's summation theorem (3.26) for 𝑎=1,𝑏=1𝑥 and 𝑐=1+𝑥.

The series (6.2) corresponds to3𝐹212,12,𝑑+1;12,𝑑(6.12) which is a special case of extended Kummer's summation theorem (5.1) for 𝑎=𝑏=1/2. Similarly the series (6.3) corresponds to3𝐹212,1𝑥,𝑑+1;12+𝑥,𝑑(6.13) which is a special case of extended Kummer's theorem (5.1) for 𝑎=1 and 𝑏=1𝑥.

The series (6.4) corresponds to 3𝐹212,12;1,𝑑+122,𝑑(6.14) which is a special case of extended Gauss's second summation theorem (5.2) for 𝑎=𝑏=1/2 or extended Bailey's summation theorem (5.3) for 𝑎=1/2, 𝑐=1.

Also, it can be easily seen that the series (6.5) to (6.8) which correspond to 4𝐹311,23,1𝑥,1+𝑑;12,,2+𝑥,𝑑4𝐹3,1,1𝑥,1+𝑥,1+𝑑;11+𝑥,2+𝑥,𝑑4𝐹331,21,1𝑥,1+𝑑;12,,1+𝑥,𝑑4𝐹312,12,145,1+𝑑;12,4,,𝑑4𝐹312,14,145,1+𝑑;14,54,,𝑑4𝐹312,12,12,,1+𝑑;12,1,𝑑4𝐹3𝑛,𝑛,𝑛,1+𝑑;11,2,𝑑(6.15) are special cases of extended Dixon's theorem (5.7).

The series (6.9) corresponds to3𝐹21,1𝑥,𝑑+1;11+𝑥,𝑑(6.16) which is a special case of (5.10) for 𝑎=1 and 𝑏=1𝑥. And the series (6.10) corresponds to4𝐹31,1𝑥,1𝑥,1+𝑑;11+𝑥,1+𝑥,𝑑(6.17) which is a special case of (5.11) for 𝑎=1, 𝑏=1𝑥=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,𝑎,𝑏,𝑑+1;12+𝑎𝑏+𝑖,𝑑3𝐹2;1𝑎,𝑏,𝑑+1212,(𝑎+𝑏+3+𝑖),𝑑3𝐹2𝑎,1𝑎+𝑖,𝑑+1;1𝑐+1,𝑑(7.1) each for 𝑖=0,±1,±2,±3,±4,±5, and4𝐹31𝑎,𝑏,𝑐,𝑑+1;12,(𝑎+𝑏+𝑖+1),2𝑐+𝑗,𝑑4𝐹3,𝑎,𝑏,𝑐,𝑑+1;12+𝑎𝑏+𝑖,1+𝑎𝑐+𝑖+𝑗,𝑑(7.2) each for 𝑖,𝑗=0,±1,±2,±3, and4𝐹3,𝑎,𝑏,𝑐,1+𝑑;1𝑒,𝑓,𝑑(7.3) where 𝑎+𝑏=1+𝑖+𝑗, 𝑒+𝑓=2𝑐+𝑗 for 𝑖,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.