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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 789519, 16 pages
Generalization of a Quadratic Transformation Formula due to Gauss
1Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Alkhodh, Muscat 123, Oman
2Department of Mathematics, Faculty of Science, Suez Canal University, Ismailia, Egypt
Received 19 April 2012; Accepted 12 June 2012
Academic Editor: Hernando Quevedo
Copyright © 2012 Medhat A. Rakha. 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.
The aim of this research paper is to obtain explicit expressions of for . For , we have the well-known transformation formula due to Gauss. The results are derived with the help of generalized Watson's theorem. Some known results obtained earlier follow special cases of our main findings.
The generalized hypergeometric function with numerator and denominator parameters is defined by [1, page 73, equation (2)] where denotes the Pochhammer symbol (or the shifted factorial, since ) defined, for any complex number , by Using the fundamental property , can be written in the form where is the well known Gamma function.
The series (1.1) is of great importance to mathematicians and physicists. All the elements , , and (similarly for (1.1)) in (1.4) are called the parameters of the series and is called the variable of the series. All four quantities , , , and may be real or complex with one exception that the denominator parameter should not be zero or a negative integer. Also it can easily been seen that if any one of the numerator parameters or or both is a negative integer, the series terminates that is, reduces to a polynomial.
The series (1.4) is known as Gauss series or the ordinary hypergeometric series and may be regarded as a generalization of the elementary geometric series. In fact (1.4) reduces to the elementary geometric series in two cases, when and and also when and .
It is interesting to mention here that in (1.4), if we replace by and let , then since we arrive at the following series: which is called the Kummer####^~^~^~^~^~^####x2019;s series or the confluent hypergeometric series.
Gauss####^~^~^~^~^~^####x2019;s hypergeometric function and its confluent case form the core of the special functions and include, as their special cases, most of the commonly elementary functions.
It should be remarked here that whenever hypergeometric and generalized hypergeometric functions reduce to gamma functions, the results are very important from an application point of view. Only a few summation theorems for the series and are available.
In this context, it is well known that the classical summation theorems such as of Gauss, Gauss second, Kummer and Bailey for the series ; Watson, Dixon, Whipple, and Saalsch####^~^~^~^~^~^####xfc;tz for the series play an important rule in the theory of hypergeometric and generalized series.
Several formulae were given by Gauss  and Kummer  expressing the product of the hypergeometric series as a hypergeometric series, such as as a series of the type and as a series of the type . In 1927, Whipple  has obtained a formula expressing as a series of the type .
By employing the above mentioned classical summation theorems for the series and , Bailey  in his well known, interesting and popular research paper made a systematic study and obtained a large number of such formulas.
Berndt  pointed out that the result (1.6) is precisely (5) of Erd####^~^~^~^~^~^####xe8;lyi treatise [9, page 111], and is the Entry 3 of the Chapter 11 of Ramanujan####^~^~^~^~^~^####x2019;s Notebooks [8, page 50] (of course, by replacing by ).
The proof of (1.7) when one of the parameters or is a negative integer was given in Watson . Subsequently, it was established more generally in the nonterminating case by Whipple . The standard proof of the nonterminating case was given in Bailey####^~^~^~^~^~^####x2019;s tract  by employing the fundamental transformation due to Thomae combined with the classical Dixon####^~^~^~^~^~^####x2019;s theorem of the sum of a . For a very recent proof of (1.7), see .
It is not out of place to mention here that in (1.6), if we replace by and let , then after a little simplification, we get the following well-known Kummer####^~^~^~^~^~^####x2019;s second theorem [4, page 140] [12, page 132], namely which also appeared as Entry 7 of the chapter 11 of Ramanujan####^~^~^~^~^~^####x2019;s Notebooks [8, page 50] (of course, by replacing by ).
Recently, a good progress has been made in generalizing the classical Watson####^~^~^~^~^~^####x2019;s theorem (1.7) on the sum of a . In 1992, Lavoie et al.  have obtained explicit expressions of
For the a recent generalization of Watson####^~^~^~^~^~^####x2019;s summation theorems and other classical summation theorems for the series and in the most general case, see .
The aim of this research paper is to obtain the explicit expressions of
In order to derive our main results, we shall require the following.(1)The following special cases of (1.13) for , recorded in : each for , , . Also, as usual, denotes the greatest integer less than or equal to , and its modulus is defined by . The coefficients and are given in Table 1.(2)The known identities [1, page 22, lemma 5; page 58, equation 1; page 52, equation 2; page 58, equation 3]
2. Main Transformation Formulae
The generalization of the quadratic transformation (1.6) due to Gauss to be established is
Also, as usual, represents the greatest integer less than or equal to , and its modulus is denoted by . The coefficients and are given in Table 1.
In order to derive our main transformation (2.1), we proceed as follows.
Proof. Denoting the left-hand side of (2.1) by , we have
Expressing as a series and after a little simplification
Using Binomial theorem (1.18), we have
which on simplification gives
Changing to and using the result [1, page 57, lemma 11]
Using (1.20) and after a little algebra
Summing up the inner series, we have separating into even and odd powers of , we have
Finally, using (1.17), (1.15), and (1.16) and after a little algebra, we easily arrive at the right-hand side of (2.1).
This completes the proof of (2.1).
3. Special Cases
In (2.1), if we put , , , we get, after summing up the series in terms of generalized hypergeometric function, the following interesting results:(i)For , (ii)For , (iii)For , (iv)For , (v)For ,
Remark 3.1. The results (3.2) and (3.3) are also recorded in .
In (3.1), (3.2), and (3.4), if we take , we get the following results:(1)For , (2)For , (3)For ,
We remark in passing that the result (3.6) is the Entry 5 of Chapter 11 in Ramanujan####^~^~^~^~^~^####x2019;s Notebooks [8, page 50] (with replaced by ), and the results (3.7) and (3.8) are closely related to (3.6).
3.1. Limiting Cases
Proof. In order to prove (4.1), we require the following result due to Kummer : Equation (4.2) is a well-known quadratic transformation recorded in Erd####^~^~^~^~^~^####xe8;lyi et al. [9, equation 4, page 111] and also recorded as an Entry 2 in the Ramanujan####^~^~^~^~^~^####x2019;s Notebooks [8, page 50]. In (4.2), if we replace by , then we have Transposing the above equation, we have Now, in (3.1) first replacing by and then replacing by and using on the right-hand side of (4.4), we get This completes the proof of (4.1).
In our next application, we would like to mention here that in 1996, there was an open problem posed by Baillon and Bruck [19, equation (5.17)] who needed to verify the following hypergeometric identity: in order to derive a quantitative form of the Ishikawa-tdelstin-####^~^~^~^~^~^####xf3; Brain asymptotic regularity theorem. Using Zeilberger####^~^~^~^~^~^####x2019;s algorithm , Baillon and Bruck  gave a computer proof of this identity which is the key to the integral representation [19, equation####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;] of their main theorem.
Soon after, Paule  gave the proof of (4.6) by using classical hypergeometric machinery by means of contiguous functions relations and Gauss####^~^~^~^~^~^####x2019;s quadratic transformation (3.1).
Our objective of this section is to obtain first three results from (3.1), (3.2), and (3.4) and then establish again three new results out of which one will be the natural generalization of the Baillon-Bruck identity (4.6).
Equation (4.7) is a natural generalization of Baillon-Bruck result (4.6). The result (4.11) is an alternate form of the Baillon-Bruck result (4.6). Its exact form can be obtained from (4.11) by using the contiguous function relation with and .
The author is highly grateful to the referee for carefully reading the manuscript and providing certain very useful suggestions which led to a better presentation of this research article. He also, would like to express his thanks to Professor A. K. Rathie (Center for Mathematical Sciences, Pala, Kerala-India) for all suggestions and for his encouraging and fruitful discussions during the preparation of this research article. The author was supported by the research Grant (IG/SCI/DOMS/12/05) funded by Sultan Qaboos University-Oman.
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