`International Journal of Mathematics and Mathematical SciencesVolumeΒ 2012Β (2012), Article IDΒ 789519, 16 pageshttp://dx.doi.org/10.1155/2012/789519`
Research Article

## 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

Copyright Β© 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.

#### Abstract

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.

#### 1. Introduction

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 special case of (1.1) for and , namely was systematically studied by Gauss [2] in 1812.

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 .

For convergence (including absolute convergence) we refer the reader to the standard texts [3] and [1].

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 [2] and Kummer [4] 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 [5] has obtained a formula expressing as a series of the type .

By employing the above mentioned classical summation theorems for the series and , Bailey [6] in his well known, interesting and popular research paper made a systematic study and obtained a large number of such formulas.

Gauss [2] obtained the following quadratic transformation formula, namely which is also contained in [7, entry (8.1.1.41), page 573].

Berndt [8] 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 ).

Bailey [6] established the result (1.6) with the help of the following classical Watson####^~^~^~^~^~^####x2019;s summation theorem [3], namely provided that .

The proof of (1.7) when one of the parameters or is a negative integer was given in Watson [10]. Subsequently, it was established more generally in the nonterminating case by Whipple [5]. The standard proof of the nonterminating case was given in Bailey####^~^~^~^~^~^####x2019;s tract [3] 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 [11].

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 ).

Very recently, Kim et al. [13] have obtained sixty six results closely related to (1.8) out of which four results are given here. These are

We remark in passing that the results (1.9) and (1.10) are also recorded in [14].

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. [15] have obtained explicit expressions of

For , we get Watson####^~^~^~^~^~^####x2019;s theorem (1.7). In the same paper [15], they have also obtained a large number of very interesting limiting and special cases of their main findings.

In [16], a summation formula for (1.7) with fixed and arbitrary was given. This result generalizes the classical Watson####^~^~^~^~^~^####x2019;s summation theorem with the case .

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 [17].

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 [15]: 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]

Table 1

#### 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.

##### 2.1. Derivation

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] we have 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 ,

Clearly, the result (3.1) is the well-known quadratic transformation due to Gauss (1.6) and the results (3.2) to (3.5) are closely related to (3.1).

Remark 3.1. The results (3.2) and (3.3) are also recorded in [18].
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

In the special cases (3.1) to (3.5), if we replace by and let , we get, after a little simplification, the known results (1.8) and (1.9) to (1.12), respectively.

#### 4. Application

In this section, we shall first establish the following result, which is given as Entry 4 in the Ramanujan####^~^~^~^~^~^####x2019;s Notebooks [8, page 50]: by employing (3.1).

Proof. In order to prove (4.1), we require the following result due to Kummer [4]: 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).

Remark 4.1. (1) The result (4.1) can also be established by employing Gauss####^~^~^~^~^~^####x2019;s summation theorem.
(2) For generalization of (4.2), see a recent paper by Kim et al. [13].

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 [20], Baillon and Bruck [19] 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 [21] 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).

For this, in our results (3.1), (3.2), and (3.4), if we replace by , we get after a little simplification the following results:

Finally, in (4.7), (4.8), and (4.9) if we take and , we get the following very interesting results:

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 .

We conclude this section by remarking that the result (4.7) is also recorded in [22] by Rathie and Kim who obtained it by other means and the results (4.8) and (4.9) are believed to be new.

#### Acknowledgments

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.

#### References

1. E. D. Rainville, Special Functions, The Macmillan, Bronx, NY, USA, 1971.
2. C. F. Gauss, βDisquisitiones generales circa seriem infinitam…,β Commentarii Societatis Regiae Scientiarum Gottingensis, vol. 2, reprinted in Werke 3, pp. 123β162, 1876.
3. W. N. Bailey, Generalized Hypergeometric Series, Cambridge university press, Cambridge, UK, 1935.
4. E. E. Kummer, βUber die hypergeometric Reihe,β Journal Für Die Reine Und Angewandte Mathematik,, vol. 1836, no. 15, pp. 127β172, 1836.
5. F. J. W. Whipple, βA group of generalized hypergeometric series: relations between 120 allied series of the type $F\left(a,b,c;e,f\right)$,β Proceedings of the London Mathematical Society, vol. 23, no. 2, pp. 104β114.
6. W. N. Bailey, βProducts of Generalized Hypergeometric Series,β Proceedings of the London Mathematical Society, vol. 28, no. 2, pp. 242β254, 1928.
7. Y. A. Brychkov, Handbook of Special Functions, CRC Press, Boca Raton, Fla, USA, 2008.
8. B. C. Berndt, Ramanujan's Notebooks. Part II, Springer, New York, NY, USA, 1989.
9. A. Erdèlyi, et al., Higher Transcendental Functions, vol. 1, Mc-Graw Hill, New York, NY, USA, 1953.
10. G. N. Watson, βA note on generalized hypergeometric series,β Proceedings of the London Mathematical Society, vol. 2, no. 23, pp. 13β15, 1925.
11. M. A. Rakha, βA new proof of the classical Watson's summation theorem,β Applied Mathematics E-Notes, vol. 11, pp. 278β282, 2011.
12. E. E. Kummer, Collected Papers, Springer, Berlin, Germany, 1975.
13. Y. S. Kim, M. A. Rakha, and A. K. Rathie, βGeneration of Kummer's second theorem with application,β Journal of Computational Mathematics and Mathematical Physics, vol. 50, no. 3, pp. 407β422, 2010.
14. A. K. Rathie and V. Nagar, βOn Kummer's second theorem involving product of generalized hypergeometric series,β Le Matematiche, vol. 50, no. 1, pp. 35β38, 1995.
15. J.-L. Lavoie, F. Grondin, and A. K. Rathie, βGeneralizations of Watson's theorem on the sum of a ${}_{3}{F}_{2}$,β Indian Journal of Mathematics, vol. 34, no. 1, pp. 23β32, 1992.
16. S. Lewanowicz, βGeneralized Watson's summation formula for ${}_{3}{F}_{2}\left(1\right)$,β Journal of Computational and Applied Mathematics, vol. 86, no. 2, pp. 375β386, 1997.
17. M. A. Rakha and A. K. Rathie, βGeneralizations of classical summation theorems for the series ${}_{2}{F}_{1}$ and ${}_{3}{F}_{2}$ with applications,β Integral Transforms and Special Functions, vol. 22, no. 11, pp. 823β840, 2011.
18. A. K. Rathie and Y. S. Kim, βOn two results contiguous to a quadratic transformation formula due to Gauss,β Far East Journal of Mathematical Sciences, vol. 3, no. 1, pp. 51β58, 2001.
19. J. Baillon and R. E. Bruck, βThe rate of asymptotic regularity is $O\left(1/\sqrt{n}\right)$,β in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 51β81, Dekker, New York, NY, USA, 1996.
20. D. Zeilberger, βA fast algorithm for proving terminating hypergeometric identities,β Discrete Mathematics, vol. 80, no. 2, pp. 207β211, 1990.
21. P. Paule, βA classical hypergeometric proof of an important transformation formula found by Baillon and Bruck,β in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 241β242, Dekker, New York, NY, USA, 1996.
22. A. K. Rathie and Y. S. Kim, βGeneralization of a transformation formula found by Baillon and Bruck,β Communicationsof the Korean Mathematical Society, vol. 23, no. 4, pp. 607β610, 2008.