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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 789519, 16 pages
http://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

Academic Editor: HernandoΒ Quevedo

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 (1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1[π‘Ž,𝑏;βˆ’4π‘₯/(1βˆ’π‘₯)22𝑏+𝑗]for 𝑗=0,Β±1,Β±2. For 𝑗=0, 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 𝑝 numerator and π‘ž denominator parameters is defined by [1, page 73, equation (2)] π‘πΉπ‘žβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π›Ό1,…,𝛼𝑝𝛽;𝑧1,…,π›½π‘ž,⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=π‘πΉπ‘žξ‚ƒπ›Ό1,…,𝛼𝑝;𝛽1,…,π›½π‘žξ‚„=;π‘§βˆžξ“π‘›=0𝛼1𝑛⋯𝛼𝑝𝑛𝛽1ξ€Έπ‘›β‹―ξ€·π›½π‘žξ€Έπ‘›π‘§π‘›,𝑛!(1.1) where (𝛼)𝑛 denotes the Pochhammer symbol (or the shifted factorial, since (1)𝑛=𝑛!) defined, for any complex number 𝛼, by (𝛼)𝑛=𝛼(𝛼+1)β‹―(𝛼+π‘›βˆ’1),π‘›βˆˆβ„•,1,𝑛=0.(1.2) Using the fundamental property Ξ“(𝛼+1)=𝛼Γ(𝛼), (𝛼)𝑛 can be written in the form (𝛼)𝑛=Ξ“(𝛼+𝑛),Ξ“(𝛼)(1.3) where Ξ“(β‹…) is the well known Gamma function.

The special case of (1.1) for 𝑝=2 and π‘ž=1, namely 2𝐹1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘Ž,𝑏;𝑧=1+π‘Žβ‹…π‘1⋅𝑐𝑧+π‘Ž(π‘Ž+1)𝑏(𝑏+1)𝑧1β‹…2𝑐(𝑐+1)2=+β‹―βˆžξ“π‘›=0(π‘Ž)𝑛(𝑏)𝑛(𝑐)𝑛𝑧𝑛,𝑛!(1.4) was systematically studied by Gauss [2] in 1812.

The series (1.1) is of great importance to mathematicians and physicists. All the elements π‘Ž, 𝑏, and 𝑐 (similarly for (1.1)) in (1.4) are called the parameters of the series and 𝑧 is called the variable of the series. All four quantities π‘Ž, 𝑏, 𝑐, and 𝑧 may be real or complex with one exception that the denominator parameter 𝑐 should not be zero or a negative integer. Also it can easily been seen that if any one of the numerator parameters π‘Ž or 𝑏 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 π‘Ž=𝑐 and 𝑏=1 and also when 𝑏=𝑐 and π‘Ž=1.

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 𝑧 by 𝑧/𝑏 and let π‘β†’βˆž, then since ((𝑏)𝑛𝑧𝑛)/𝑏𝑛→𝑧𝑛 we arrive at the following series: 1𝐹1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘Žπ‘βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘Ž;𝑧=1+1⋅𝑐𝑧+π‘Ž(π‘Ž+1)𝑧1β‹…2𝑐(𝑐+1)2=+β‹―βˆžξ“π‘›=0(π‘Ž)𝑛(𝑐)𝑛𝑧𝑛,𝑛!(1.5) which is called the Kummer’s series or the confluent hypergeometric series.

Gauss’s hypergeometric function 2𝐹1 and its confluent case 1𝐹1 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 2𝐹1 and 3𝐹2 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 2𝐹1; Watson, Dixon, Whipple, and Saalschütz for the series 3𝐹2 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 𝑒1βˆ’π‘₯𝐹1(π‘₯) as a series of the type 1𝐹1(βˆ’π‘₯) and (1+π‘₯)2βˆ’π‘πΉ1[4π‘₯/(1+π‘₯)2] as a series of the type 2𝐹1(π‘₯). In 1927, Whipple [5] has obtained a formula expressing (1βˆ’π‘₯)3βˆ’π‘πΉ2[βˆ’4π‘₯/(1βˆ’π‘₯)2] as a series of the type 3𝐹2.

By employing the above mentioned classical summation theorems for the series 2𝐹1 and 3𝐹2, 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 (1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž,𝑏;βˆ’4π‘₯(1βˆ’π‘₯)2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=2𝑏2𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž,π‘Žβˆ’π‘+2;π‘₯21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,(1.6) 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 ErdΓ¨lyi treatise [9, page 111], and is the Entry 3 of the Chapter 11 of Ramanujan’s Notebooks [8, page 50] (of course, by replacing π‘₯ by βˆ’π‘₯).

Bailey [6] established the result (1.6) with the help of the following classical Watson’s summation theorem [3], namely 3𝐹2⎑⎒⎒⎒⎒⎣1π‘Ž,𝑏,𝑐;12⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=(π‘Ž+𝑏+1),2𝑐Γ(1/2)Ξ“(𝑐+1/2)Ξ“(π‘βˆ’(1/2)π‘Žβˆ’(1/2)𝑏+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)Ξ“(π‘βˆ’(1/2)π‘Ž+1/2)Ξ“(π‘βˆ’(1/2)𝑏+1/2)(1.7) provided that Re(2π‘βˆ’π‘Žβˆ’π‘)>βˆ’1.

The proof of (1.7) when one of the parameters π‘Ž or 𝑏 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’s tract [3] by employing the fundamental transformation due to Thomae combined with the classical Dixon’s theorem of the sum of a 3𝐹2. 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 π‘₯ by π‘₯/π‘Ž and let π‘Žβ†’βˆž, then after a little simplification, we get the following well-known Kummer’s second theorem [4, page 140] [12, page 132], namely 𝑒1βˆ’π‘₯/2𝐹1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦=;π‘₯2𝑏0𝐹1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’;π‘₯2116𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,(1.8) which also appeared as Entry 7 of the chapter 11 of Ramanujan’s Notebooks [8, page 50] (of course, by replacing π‘₯ by π‘₯/2).

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 𝑒1βˆ’π‘₯/2𝐹1(𝛼;2𝛼+1;π‘₯)=0𝐹1ξ‚΅1βˆ’;𝛼+2;π‘₯2ξ‚Άβˆ’π‘₯162(2𝛼+1)0𝐹1ξ‚΅3βˆ’;𝛼+2;π‘₯2𝑒16,(1.9)1βˆ’π‘₯/2𝐹1(𝛼;2π›Όβˆ’1;π‘₯)=0𝐹1ξ‚΅1βˆ’;π›Όβˆ’2;π‘₯2ξ‚Ά+π‘₯162(2π›Όβˆ’1)0𝐹1ξ‚΅1βˆ’;𝛼+2;π‘₯2𝑒16,(1.10)1βˆ’π‘₯/2𝐹1(𝛼;2𝛼+2;π‘₯)=0𝐹1ξ‚΅3βˆ’;𝛼+2;π‘₯2ξ‚Άβˆ’π‘₯162(𝛼+1)0𝐹1ξ‚΅3βˆ’;𝛼+2;π‘₯2ξ‚Ά+π‘₯1624(𝛼+1)(2𝛼+3)0𝐹1ξ‚΅5βˆ’;𝛼+2;π‘₯2ξ‚Ά,𝑒16(1.11)1βˆ’π‘₯/2𝐹1(𝛼;2π›Όβˆ’2;π‘₯)=0𝐹1ξ‚΅1βˆ’;π›Όβˆ’2;π‘₯2ξ‚Ά+π‘₯162(π›Όβˆ’1)0𝐹1ξ‚΅3βˆ’;π›Όβˆ’2;π‘₯2ξ‚Ά+π‘₯1624(π›Όβˆ’1)(2π›Όβˆ’1)0𝐹1ξ‚΅1βˆ’;𝛼+2;π‘₯2ξ‚Ά.16(1.12)

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’s theorem (1.7) on the sum of a 3𝐹2. In 1992, Lavoie et al. [15] have obtained explicit expressions of 3𝐹2⎑⎒⎒⎒⎒⎣1π‘Ž,𝑏,𝑐;12⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦(π‘Ž+𝑏+𝑖+1),2𝑐+𝑗for𝑖,𝑗=0,Β±1,Β±2.(1.13)

For 𝑖=𝑗=0, we get Watson’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 𝑗 and arbitrary 𝑖(𝑖,π‘—βˆˆβ„€) was given. This result generalizes the classical Watson’s summation theorem with the case 𝑖=𝑗=0.

For the a recent generalization of Watson’s summation theorems and other classical summation theorems for the series 2𝐹1 and 3𝐹2 in the most general case, see [17].

The aim of this research paper is to obtain the explicit expressions of (1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž,𝑏;βˆ’4π‘₯(1βˆ’π‘₯)2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦2𝑏+𝑗for𝑗=0,Β±1,Β±2.(1.14)

In order to derive our main results, we shall require the following.(1)The following special cases of (1.13) for 𝑖=0, recorded in [15]: 3𝐹2⎑⎒⎒⎒⎒⎣1βˆ’2𝑛,2π‘Ž+2𝑛,𝑐;1π‘Ž+2⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,2𝑐+𝑗=𝐷𝑗(1/2)𝑛([])π‘Žβˆ’π‘+(1/2)βˆ’π‘—/2𝑛(π‘Ž+(1/2))𝑛[])(𝑐+(1/2)+𝑗/2𝑛,(1.15)3𝐹2⎑⎒⎒⎒⎒⎣1βˆ’2π‘›βˆ’1,2π‘Ž+2𝑛+1,𝑐;1π‘Ž+2⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,2𝑐+𝑗=𝐸𝑗(3/2)𝑛[])(π‘Žβˆ’π‘+(3/2)βˆ’(𝑗+1)/2𝑛(π‘Ž+(1/2))𝑛[])(𝑐+(1/2)+(𝑗+1)/2𝑛(1.16) each for 𝑗=0, Β±1, Β±2. Also, as usual, [π‘₯] denotes the greatest integer less than or equal to π‘₯, and its modulus is defined by |π‘₯|. The coefficients 𝐷𝑗 and 𝐸𝑗 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] (πœ†)2𝑛=22𝑛12πœ†ξ‚π‘›ξ‚€121πœ†+2𝑛((π‘›βˆˆβ„•βˆͺ{0}),(1.17)1βˆ’π‘₯)βˆ’π‘Ž=βˆžξ“π‘›=0(π‘Ž)𝑛π‘₯𝑛!𝑛,(1.18)(πœ†)π‘›βˆ’π‘˜=(βˆ’1)π‘˜(πœ†)𝑛(1βˆ’πœ†βˆ’π‘›)π‘˜(0β‰€π‘˜β‰€π‘›;π‘›βˆˆβ„•βˆͺ{0}),(1.19)(π‘›βˆ’π‘˜)!=(βˆ’1)π‘˜π‘›!(βˆ’π‘›)π‘˜(0β‰€π‘˜β‰€π‘›;π‘›βˆˆβ„•βˆͺ{0}).(1.20)

tab1
Table 1

2. Main Transformation Formulae

The generalization of the quadratic transformation (1.6) due to Gauss to be established is (1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž,𝑏;βˆ’4π‘₯(1βˆ’π‘₯)2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=2𝑏+π‘—βˆžξ“π‘›=0𝐷𝑗(π‘Ž)𝑛[])(π‘Žβˆ’π‘+(1/2)βˆ’π‘—/2𝑛[])(𝑏+(1/2)+𝑗/2𝑛π‘₯𝑛!2𝑛+2π‘Žβˆžξ“π‘›=0𝐸𝑗(π‘Ž+1)𝑛([(])π‘Žβˆ’π‘+(3/2)βˆ’π‘—+1)/2𝑛[])(𝑏+(1/2)+(𝑗+1)/2𝑛π‘₯𝑛!2𝑛+1for𝑗=0,Β±1,Β±2.(2.1)

Also, as usual, [π‘₯] represents the greatest integer less than or equal to π‘₯, and its modulus is denoted by |π‘₯|. The coefficients 𝐷𝑗 and 𝐸𝑗 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 𝑆, we have 𝑆=(1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž,𝑏;βˆ’4π‘₯(1βˆ’π‘₯)2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦2𝑏+𝑗.(2.2) Expressing 2𝐹1 as a series and after a little simplification =βˆžξ“π‘˜=0(π‘Ž)π‘˜(𝑏)π‘˜(βˆ’1)π‘˜22π‘˜π‘₯π‘˜π‘˜!(2𝑏+𝑗)π‘˜(1βˆ’π‘₯)βˆ’(2π‘Ž+2π‘˜).(2.3) Using Binomial theorem (1.18), we have =βˆžξ“π‘˜=0(π‘Ž)π‘˜(𝑏)π‘˜(βˆ’1)π‘˜22π‘˜π‘₯π‘˜π‘˜!(2𝑏+𝑗)π‘˜βˆžξ“π‘›=0(2π‘Ž+2π‘˜)𝑛π‘₯𝑛!𝑛,(2.4) which on simplification gives =βˆžξ“βˆžπ‘˜=0𝑛=0(2π‘Ž+2π‘˜)𝑛(π‘Ž)π‘˜(𝑏)π‘˜(βˆ’1)π‘˜22π‘˜(2𝑏+𝑗)π‘˜π‘₯𝑛!π‘˜!𝑛+π‘˜.(2.5) Changing 𝑛 to π‘›βˆ’π‘˜ and using the result [1, page 57, lemma 11] βˆžξ“βˆžπ‘›=0ξ“π‘˜=0𝐴(π‘˜,𝑛)=βˆžξ“π‘›π‘›=0ξ“π‘˜=0𝐴(π‘˜,π‘›βˆ’π‘˜),(2.6) we have 𝑆=βˆžξ“π‘›π‘›=0ξ“π‘˜=0(2π‘Ž+2π‘˜)π‘›βˆ’π‘˜(π‘Ž)π‘˜(𝑏)π‘˜(βˆ’1)π‘˜22π‘˜(2𝑏+𝑗)π‘˜π‘₯(π‘›βˆ’π‘˜)!π‘˜!𝑛=βˆžξ“π‘›π‘›=0ξ“π‘˜=0Ξ“(2π‘Ž+𝑛+π‘˜)Ξ“(2π‘Ž+2π‘˜)(π‘Ž)π‘˜(𝑏)π‘˜(βˆ’1)π‘˜22π‘˜(2𝑏+𝑗)π‘˜(π‘₯π‘›βˆ’π‘˜)!π‘˜!𝑛.(2.7) Using (1.20) and after a little algebra =βˆžξ“π‘›=0(2π‘Ž)𝑛π‘₯𝑛!π‘›π‘›ξ“π‘˜=0(βˆ’π‘›)π‘˜(2π‘Ž+𝑛)π‘˜(𝑏)π‘˜(π‘Ž+(1/2))π‘˜(2𝑏+𝑗)π‘˜.π‘˜!(2.8)
Summing up the inner series, we have 𝑆=βˆžξ“π‘›=0(2π‘Ž)𝑛π‘₯𝑛!𝑛3𝐹2⎑⎒⎒⎒⎒⎣1βˆ’π‘›,2π‘Ž+𝑛,𝑏;1π‘Ž+2⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,2𝑏+𝑗,(2.9) separating into even and odd powers of π‘₯, we have 𝑆=βˆžξ“π‘›=0(2π‘Ž)2𝑛π‘₯(2𝑛)!32𝑛𝐹2⎑⎒⎒⎒⎒⎣1βˆ’2𝑛,2π‘Ž+2𝑛,𝑏;1π‘Ž+2⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦+,2𝑏+π‘—βˆžξ“π‘›=0(2π‘Ž)2𝑛+1π‘₯(2𝑛+1)!32𝑛+1𝐹2⎑⎒⎒⎒⎒⎣1βˆ’2π‘›βˆ’1,2π‘Ž+2𝑛+1,𝑏;1π‘Ž+2⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.,2𝑏+𝑗(2.10)
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 𝑗=0, Β±1, Β±2, we get, after summing up the series in terms of generalized hypergeometric function, the following interesting results:(i)For 𝑗=0, (1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž,𝑏;βˆ’4π‘₯(1βˆ’π‘₯)2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=2𝑏2𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž,π‘Žβˆ’π‘+2;π‘₯21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(3.1)(ii)For 𝑗=1, (1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž,𝑏;βˆ’4π‘₯(1βˆ’π‘₯)2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=2𝑏+12𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž,π‘Žβˆ’π‘+2;π‘₯21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+2π‘Žπ‘₯(2𝑏+1)2𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž+1,π‘Žβˆ’π‘+2;π‘₯23𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(3.2)(iii)For 𝑗=βˆ’1, (1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž,𝑏;βˆ’4π‘₯(1βˆ’π‘₯)2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=2π‘βˆ’12𝐹1⎑⎒⎒⎒⎒⎒⎣3π‘Ž,π‘Žβˆ’π‘+2;π‘₯21π‘βˆ’2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’2π‘Žπ‘₯(2π‘βˆ’1)2𝐹1⎑⎒⎒⎒⎒⎒⎣3π‘Ž+1,π‘Žβˆ’π‘+2;π‘₯21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(3.3)(iv)For 𝑗=2, (1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž,𝑏;βˆ’4π‘₯(1βˆ’π‘₯)2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=2𝑏+22𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž,π‘Žβˆ’π‘+2;π‘₯23𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+2π‘Žπ‘₯(𝑏+1)2𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž+1,π‘Žβˆ’π‘+2;π‘₯23𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+4π‘Ž(π‘Ž+1)π‘₯(𝑏+1)(2𝑏+3)22𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž+2,π‘Žβˆ’π‘+2;π‘₯25𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(3.4)(v)For 𝑗=βˆ’2, (1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž,𝑏;βˆ’4π‘₯(1βˆ’π‘₯)2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=2π‘βˆ’22𝐹1⎑⎒⎒⎒⎒⎒⎣3π‘Ž,π‘Žβˆ’π‘+2;π‘₯21π‘βˆ’2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’2π‘Žπ‘₯(π‘βˆ’1)2𝐹1⎑⎒⎒⎒⎒⎒⎣5π‘Ž+1,π‘Žβˆ’π‘+2;π‘₯21π‘βˆ’2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+4π‘Ž(π‘Ž+1)π‘₯(π‘βˆ’1)(2π‘βˆ’1)22𝐹1⎑⎒⎒⎒⎒⎒⎣5π‘Ž+2,π‘Žβˆ’π‘+2;π‘₯21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(3.5)

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 𝑏=1/2, we get the following results:(1)For 𝑗=0, (1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1⎑⎒⎒⎒⎒⎒⎣1π‘Ž,2;βˆ’4π‘₯(1βˆ’π‘₯)21⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=2𝐹1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž,π‘Ž;π‘₯21⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.(3.6)(2)For 𝑗=1, (1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1⎑⎒⎒⎒⎒⎒⎣1π‘Ž,2;βˆ’4π‘₯(1βˆ’π‘₯)22⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=2𝐹1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž,π‘Ž;π‘₯21⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦+π‘Žπ‘₯2𝐹1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž+1,π‘Ž;π‘₯22⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.(3.7)(3)For 𝑗=2, (1βˆ’π‘₯)2βˆ’2π‘ŽπΉ1⎑⎒⎒⎒⎒⎒⎣1π‘Ž,2;βˆ’4π‘₯(1βˆ’π‘₯)23⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=2𝐹1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž,π‘Ž;π‘₯22⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦+4π‘Žπ‘₯32𝐹1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž+1,π‘Ž;π‘₯22⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦+2π‘Ž(π‘Ž+1)3π‘₯22𝐹1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘Ž+2,π‘Ž;π‘₯23⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.(3.8)
We remark in passing that the result (3.6) is the Entry 5 of Chapter 11 in Ramanujan’s Notebooks [8, page 50] (with π‘₯ replaced by βˆ’π‘₯), 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 π‘₯ by π‘₯/𝑏 and let π‘β†’βˆž, 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’s Notebooks [8, page 50]: 2𝐹1⎑⎒⎒⎒⎒⎒⎣121π‘Ž,21π‘Ž+2;4π‘₯(1+π‘₯)21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=(1+π‘₯)π‘Ž2𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž,π‘Žβˆ’π‘+21;π‘₯𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,(4.1) by employing (3.1).

Proof. In order to prove (4.1), we require the following result due to Kummer [4]: 2𝐹1⎑⎒⎒⎒⎒⎣;π‘Ž,𝑏2π‘₯⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=1+π‘₯2𝑏(1+π‘₯)π‘Ž2𝐹1⎑⎒⎒⎒⎒⎒⎣121π‘Ž,21π‘Ž+2;π‘₯21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(4.2) Equation (4.2) is a well-known quadratic transformation recorded in ErdΓ¨lyi et al. [9, equation 4, page 111] and also recorded as an Entry 2 in the Ramanujan’s Notebooks [8, page 50]. In (4.2), if we replace π‘₯ by 2√π‘₯/(1+π‘₯), then we have 2𝐹1⎑⎒⎒⎒⎒⎒⎒⎣;4βˆšπ‘Ž,𝑏π‘₯ξ‚€βˆš1+π‘₯2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=ξ‚€βˆš2𝑏1+π‘₯2π‘Ž(1+π‘₯)π‘Ž2𝐹1⎑⎒⎒⎒⎒⎒⎣121π‘Ž,21π‘Ž+2;4π‘₯(1+π‘₯)21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(4.3) Transposing the above equation, we have 2𝐹1⎑⎒⎒⎒⎒⎒⎣121π‘Ž,21π‘Ž+2;4π‘₯(1+π‘₯)21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=(1+π‘₯)π‘Žξ‚€βˆš1+π‘₯22π‘ŽπΉ1⎑⎒⎒⎒⎒⎒⎒⎣;4βˆšπ‘Ž,𝑏π‘₯ξ‚€βˆš1+π‘₯2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦2𝑏.(4.4) Now, in (3.1) first replacing π‘₯ by βˆ’π‘₯ and then replacing π‘₯ by √π‘₯ and using on the right-hand side of (4.4), we get 2𝐹1⎑⎒⎒⎒⎒⎒⎣121π‘Ž,21π‘Ž+2;4π‘₯(1+π‘₯)21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=(1+π‘₯)π‘Ž2𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž,π‘Žβˆ’π‘+21;π‘₯𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(4.5) This completes the proof of (4.1).

Remark 4.1. (1) The result (4.1) can also be established by employing Gauss’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: 2𝐹1⎑⎒⎒⎒⎒⎣1βˆ’π‘š,22⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦;4𝑧(1βˆ’π‘§)=(π‘š+1)(1βˆ’π‘§)𝑧22π‘šβˆ’1𝐹1⎑⎒⎒⎒⎒⎣;ξ‚€βˆ’π‘š,βˆ’π‘š1βˆ’π‘§π‘§ξ‚22⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦+(2π‘§βˆ’1)𝑧22π‘šβˆ’1𝐹1⎑⎒⎒⎒⎒⎣;ξ‚€βˆ’π‘š,βˆ’π‘š1βˆ’π‘§π‘§ξ‚21⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,(4.6) in order to derive a quantitative form of the Ishikawa-tdelstin-Γ³ Brain asymptotic regularity theorem. Using Zeilberger’s algorithm [20], Baillon and Bruck [19] gave a computer proof of this identity which is the key to the integral representation [19, equation  (2.1)] 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’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 π‘₯ by (π‘§βˆ’1)/𝑧, we get after a little simplification the following results: 2𝐹1⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘Ž,𝑏;4𝑧(1βˆ’π‘§)2𝑏=𝑧2βˆ’2π‘ŽπΉ1⎑⎒⎒⎒⎒⎒⎣1π‘Ž,π‘Žβˆ’π‘+2;ξ‚€1βˆ’π‘§π‘§ξ‚21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,(4.7)2𝐹1⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘Ž,𝑏;4𝑧(1βˆ’π‘§)2𝑏+1=π‘§βˆ’2π‘ŽβŽ§βŽͺβŽͺ⎨βŽͺβŽͺ⎩2𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž,π‘Žβˆ’π‘+2;ξ‚€1βˆ’π‘§π‘§ξ‚21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’2π‘Žξ‚€2𝑏+11βˆ’π‘§π‘§ξ‚2𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž+1,π‘Žβˆ’π‘+2;ξ‚€1βˆ’π‘§π‘§ξ‚23𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦⎫βŽͺβŽͺ⎬βŽͺβŽͺ⎭,(4.8)2𝐹1⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘Ž,𝑏;4𝑧(1βˆ’π‘§)2𝑏+2=π‘§βˆ’2π‘ŽβŽ§βŽͺβŽͺ⎨βŽͺβŽͺ⎩2𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž,π‘Žβˆ’π‘+2;ξ‚€1βˆ’π‘§π‘§ξ‚21𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’2π‘Žξ‚€2𝑏+11βˆ’π‘§π‘§ξ‚2𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž+1,π‘Žβˆ’π‘+2;ξ‚€1βˆ’π‘§π‘§ξ‚23𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+4π‘Ž(π‘Ž+1)ξ‚€(𝑏+1)(2𝑏+3)1βˆ’π‘§π‘§ξ‚22𝐹1⎑⎒⎒⎒⎒⎒⎣1π‘Ž+2,π‘Žβˆ’π‘+2;ξ‚€1βˆ’π‘§π‘§ξ‚25𝑏+2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦⎫βŽͺβŽͺ⎬βŽͺβŽͺ⎭.(4.9)

Finally, in (4.7), (4.8), and (4.9) if we take π‘Ž=βˆ’π‘š and 𝑏=1/2, we get the following very interesting results: 2𝐹1⎑⎒⎒⎒⎒⎣121⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,βˆ’π‘š;4𝑧(1βˆ’π‘§)=𝑧22π‘šπΉ1⎑⎒⎒⎒⎒⎣;ξ‚€βˆ’π‘š,βˆ’π‘š1βˆ’π‘§π‘§ξ‚21⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,(4.10)2𝐹1⎑⎒⎒⎒⎒⎣121⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,βˆ’π‘š;4𝑧(1βˆ’π‘§)=𝑧2π‘šβŽ§βŽͺβŽͺ⎨βŽͺβŽͺ⎩2𝐹1⎑⎒⎒⎒⎒⎣;ξ‚€βˆ’π‘š,βˆ’π‘š1βˆ’π‘§π‘§ξ‚21⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ‚€+π‘š1βˆ’π‘§π‘§ξ‚2𝐹1⎑⎒⎒⎒⎒⎣;ξ‚€βˆ’π‘š,βˆ’π‘š+11βˆ’π‘§π‘§ξ‚22⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎫βŽͺβŽͺ⎬βŽͺβŽͺ⎭,(4.11)2𝐹1⎑⎒⎒⎒⎒⎣122⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,βˆ’π‘š;4𝑧(1βˆ’π‘§)=𝑧2π‘šβŽ§βŽͺβŽͺ⎨βŽͺβŽͺ⎩2𝐹1⎑⎒⎒⎒⎒⎣;ξ‚€βˆ’π‘š,βˆ’π‘š1βˆ’π‘§π‘§ξ‚21⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦+4π‘š3ξ‚€1βˆ’π‘§π‘§ξ‚2𝐹1⎑⎒⎒⎒⎒⎣;ξ‚€βˆ’π‘š,βˆ’π‘š+11βˆ’π‘§π‘§ξ‚22⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’2π‘š(1βˆ’π‘š)3ξ‚€1βˆ’π‘§π‘§ξ‚22𝐹1⎑⎒⎒⎒⎒⎣;ξ‚€βˆ’π‘š,βˆ’π‘š+21βˆ’π‘§π‘§ξ‚23⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎫βŽͺβŽͺ⎬βŽͺβŽͺ⎭.(4.12)

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 π‘Žπ‘π‘§π‘(π‘βˆ’1)2𝐹1⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=π‘Ž+1,𝑏+1;𝑧𝑐+12𝐹1⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’π‘Ž,𝑏;𝑧𝑐+12𝐹1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘Ž,𝑏;𝑧(4.13) with π‘Ž=𝑏=βˆ’π‘š and 𝑐=1.

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.

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