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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 309503, 27 pages
Extensions of Certain Classical Summation Theorems for the Series , , and 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 © 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.
Motivated by the extension of classical Gauss's summation theorem for the series 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 , Watson, Dixon and Whipple for the series , and a few other hypergeometric identities for the series and . 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.
In 1812, Gauss  systematically discussed the series where denotes the Pochhammer symbol defined (for ) by It is noted that the series (1.1) and its natural generalization 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 and also when and . The series (1.1) is represented by the notation or which is usually referred to as Gauss hypergeometric function. In (1.1), the three elements , , and are described as the parameters of the series, and is called the variable of the series. All four of these quantities may be real or complex with an exception that is neither zero nor a negative integer. Also, in (1.1), it is easy to see that if any one of the numerator parameters or 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 provided that . Also when , the series is absolutely convergent if , conditionally convergent if , and divergent if .
Further, if in (1.1), we replace by and let , then , and we arrive to the following Kummer's series
This series is absolutely convergent for all values of , , and , real or complex, excluding and is represented by the notation or which is called a confluent hypergeometric function.
Gauss hypergeometric function and its confluent case form the core special functions and include, as their special cases, most of the commonly used functions. Thus 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 and 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 is the generalized hypergeometric series defined by
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 , and Watson, Dixon, and Whipple for the series play an important role in the theory of generalized hypergeometric series. It has been pointed out by Berndt , 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 , a large number of very interesting results involving products of generalized hypergeometric series have been developed. In  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 and Watson, Dixon, and Whipple for the series 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 and . These are the following.
Gauss theorem : provided .
Kummer theorem :
Gauss’s second theorem :
Bailey theorem :
Watson theorem : provided .
Dixon theorem : provided .
Whipple theorem : provided and with and .
Other hypergeometric identities : provided .
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  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 .(i)For ,(ii)For ,(iii)For ,(iv)For ,(v)For ,(vi)For ,(vii)For ,(viii)For , We now come to the derivations of these summation in brief.
The series (3.2) corresponds to which is a special case of Kummer's summation theorem (2.2) for . Similarly the series (3.3) corresponds to which is a special case of Kummer's summation theorem (2.2) for , .
Also, it can easily be seen that the series (3.5) to (3.9) correspond to each of the following series: which are special cases of classical Dixon's theorem (2.6) for (i) , , , (ii) , , (iii) , , , (iv) , , (v) , , (vi) , and (vii) , respectively.
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 ). 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 of each for , and for for ; , and for , for .
Notice that, if we denote (3.23) by , the natural symmetry makes it possible to extend the result to .
On the other hand the following very interesting result for the series (written here in a slightly different form) is given in the literature (e.g., see ) provided and .
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 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 :(ii)Contiguous Gauss's Second theorem :(iii)Contiguous Bailey's theorem :(iv)Contiguous Watson's theorem : provided that . provided that . provided that .(v)Contiguous Dixon's theorem : provided that . provided that . provided that .(vi)Contiguous Whipple's theorem : provided that . provided that .
5. Main Summation Formulas
In this section, the following extensions of the classical summation theorems will be established. In all these theorems we have .(i)Extension of Kummer's theorem:(ii)Extension of Gauss's second theorem:(iii)Extension of Bailey's theorem:(iv)Extension of Watson's theorem:
First Extension: provided .
Second Extension: provided , and are given by (v)Extension of Dixon's theorem: provided , and are given by (vi)Extension of Whipple's theorem: provided .(vii)Extension of (2.8):(viii)Extension of (2.9): provided .
In order to derive (5.1), it is just a simple exercise to prove the following relation:
Now, it is easy to see that the first and second 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: 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 , we get Kummer's theorem (2.2).(2)In (5.2), if we take , we get Gauss's second theorem (2.3).(3)In (5.3), if we take , we get Bailey's theorem (2.4).(4)In (5.4), if we take , we get Watson's theorem (2.5).(5)In (5.5), if we take , we again get Watson's theorem (2.5).(6)In (5.7), if we take , we get Dixon's theorem (2.6).(7)In (5.9), if we take , we get Whipple's theorem (2.7).(8)In (5.10), if we take , we get (2.8).(9)In (5.11), if we take , we get (2.9).
6. Generalizations of Summations Due to Ramanujan
In all the summations, we have .(i)For :(ii)For :(iii)For :(iv)For :(v)For :(vi)For :(vii)For :(viii)For :
The series (6.2) corresponds to which is a special case of extended Kummer's summation theorem (5.1) for . Similarly the series (6.3) corresponds to which is a special case of extended Kummer's theorem (5.1) for and .
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
each for , and each for , and where , for are also under investigations and will be published later.
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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