Generalization of a Quadratic Transformation Formula due to Gauss

Abstract

The aim of this research paper is to obtain explicit expressions of (1−𝑥)^2−2𝑎𝐹₁[^{𝑎,𝑏;−4𝑥/(1−𝑥)2}_2𝑏+𝑗]for 𝑗=0,±1,±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 𝑞 denominator parameters is defined by [1, page 73, equation (2)] _𝑝𝐹_𝑞⎡⎢⎢⎢⎢⎣𝛼₁,…,𝛼_𝑝𝛽;𝑧₁,…,𝛽_𝑞,⎤⎥⎥⎥⎥⎦=^𝑝=𝐹_𝑞𝛼₁,…,𝛼_𝑝;𝛽₁,…,𝛽_𝑞;𝑧_∞_𝑛=0𝛼₁_𝑛⋯𝛼_𝑝_𝑛𝛽₁_𝑛⋯𝛽_𝑞_𝑛𝑧^𝑛𝑛!,(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=𝛼1,(𝛼+1)⋯(𝛼+𝑛−1),𝑛𝑛=∈0ℕ.,(1.2) Using the fundamental property Γ(𝛼+1)=𝛼Γ(𝛼), (𝛼)_𝑛 can be written in the form (𝛼)_𝑛=Γ(𝛼+𝑛)Γ(𝛼),(1.3) where Γ(⋅) is the well known Gamma function.

The special case of (1.1) for 𝑝=2 and 𝑞=1, namely ₂𝐹₁⎡⎢⎢⎢⎢⎣𝑎𝑐,𝑏;𝑧⎤⎥⎥⎥⎥⎦=1+𝑎⋅𝑏1⋅𝑐𝑧+𝑎(1𝑎⋅+21𝑐)(𝑏𝑐(+𝑏1+)1)𝑧²=+⋯_∞_𝑛=0(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛𝑧^𝑛𝑛!,(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 𝑎, 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 𝑎, 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 𝑎 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 𝑎=𝑐 and 𝑏=1 and also when 𝑏=𝑐 and 𝑎=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 𝑏→∞, then since ((𝑏)_𝑛𝑧^𝑛)/𝑏^𝑛→𝑧^𝑛 we arrive at the following series: ₁𝐹₁⎡⎢⎢⎢⎢⎣𝑎𝑐;𝑧⎤⎥⎥⎥⎥⎦=1+𝑎1⋅𝑐𝑧+1⋅𝑎2(𝑐𝑎(+𝑐+1)1)𝑧²=+⋯_∞_𝑛=0(𝑎)_𝑛(𝑐)_𝑛𝑧^𝑛𝑛!,(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 𝑒^1−𝑥𝐹₁(𝑥) as a series of the type ₁𝐹₁(−𝑥) and (1+𝑥)^2−𝑝𝐹₁[4𝑥/(1+𝑥)²] as a series of the type ₂𝐹₁(𝑥). In 1927, Whipple [5] has obtained a formula expressing (1−𝑥)^3−𝑝𝐹₂[−4𝑥/(1−𝑥)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 (1−𝑥)^2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑏;−4𝑥(1−𝑥)²2𝑏⎤⎥⎥⎥⎥⎥⎦=²𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑎−𝑏+12;𝑥₂𝑏+12⎤⎥⎥⎥⎥⎥⎦,(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 Erdèlyi treatise [9, page 111], and is the Entry 3 of the Chapter 11 of Ramanujan’s Notebooks [8, page 50] (of course, by replacing 𝑥 by −𝑥).

Bailey [6] established the result (1.6) with the help of the following classical Watson’s summation theorem [3], namely ₃𝐹₂⎡⎢⎢⎢⎢⎣1𝑎,𝑏,𝑐;1=2(Γ𝑎(+1/2𝑏)+Γ1(𝑐),+21𝑐/2)Γ(𝑐⎤⎥⎥⎥⎥⎦−(1/2)𝑎−(1/2)𝑏+1/2)Γ((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),(1.7) provided that Re(2𝑐−𝑎−𝑏)>−1.

The proof of (1.7) when one of the parameters 𝑎 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 𝑥 by 𝑥/𝑎 and let 𝑎→∞, then after a little simplification, we get the following well-known Kummer’s second theorem [4, page 140] [12, page 132], namely 𝑒^1−𝑥/2𝐹₁⎡⎢⎢⎢⎢⎣2𝑏𝑏;𝑥⎤⎥⎥⎥⎥⎦=₀𝐹₁⎡⎢⎢⎢⎢⎢⎣−;𝑥₂𝑏+1162⎤⎥⎥⎥⎥⎥⎦,(1.8) which also appeared as Entry 7 of the chapter 11 of Ramanujan’s Notebooks [8, page 50] (of course, by replacing 𝑥 by 𝑥/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 𝑒^1−𝑥/2𝐹₁(𝛼;2𝛼+1;𝑥)=₀𝐹₁−;𝛼+12;1𝑥6²−2(2𝛼𝑥+1)⁰𝐹₁𝑒−;𝛼+32;1𝑥6²,(1.9)_1−𝑥/2𝐹₁(𝛼;2𝛼−1;𝑥)=₀𝐹₁−;𝛼−12;1𝑥6²+2(2𝛼𝑥−1)⁰𝐹₁𝑒−;𝛼+12;1𝑥6²,(1.10)_1−𝑥/2𝐹₁(𝛼;2𝛼+2;𝑥)=₀𝐹₁−;𝛼+32;1𝑥6²−2(𝛼𝑥+1)⁰𝐹₁+𝑥−;𝛼+32;1𝑥6²₂4(𝛼+1)(2𝛼+3)⁰𝐹₁𝑒−;𝛼+25;1𝑥6²,(1.11)_1−𝑥/2𝐹₁(𝛼;2𝛼−2;𝑥)=₀𝐹₁−;𝛼−12;1𝑥6²+2(𝛼𝑥−1)⁰𝐹₁+𝑥−;𝛼−32;1𝑥6²₂4(𝛼−1)(2𝛼−1)⁰𝐹₁−;𝛼+21;1𝑥6².(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⎡⎢⎢⎢⎢⎣1𝑎,𝑏,𝑐;12(𝑎+𝑏+𝑖+1),2𝑐+𝑗⎤⎥⎥⎥⎥⎦for𝑖,𝑗=0,±1,±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,𝑗∈ℤ) 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 (1−𝑥)^2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑏;−4𝑥(1−𝑥)²2𝑏+𝑗⎤⎥⎥⎥⎥⎥⎦for𝑗=0,±1,±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⎡⎢⎢⎢⎢⎣𝑎−+2𝑛1,2𝑎+2𝑛,𝑐;12,2𝑐+𝑗⎤⎥⎥⎥⎥⎦=𝐷^𝑗(1/2)_𝑛(𝑎−𝑐+(1/2)−[𝑗/2])_𝑛(𝑎+(1/2))_𝑛(𝑐+(1/2)+[𝑗/2])_𝑛,(1.15)₃𝐹₂⎡⎢⎢⎢⎢⎣−𝑎2𝑛+−11,2𝑎+2𝑛+1,𝑐;12,2𝑐+𝑗⎤⎥⎥⎥⎥⎦=𝐸^𝑗(3/2)_𝑛(𝑎−𝑐+(3/2)−[(𝑗+1)/2])_𝑛(𝑎+(1/2))_𝑛(𝑐+(1/2)+[(𝑗+1)/2])_𝑛(1.16) each for 𝑗=0, ±1, ±2. Also, as usual, [𝑥] denotes the greatest integer less than or equal to 𝑥, and its modulus is defined by |𝑥|. The coefficients 𝐷_𝑗 and 𝐸_𝑗 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] (𝜆)_2𝑛=2^2𝑛12𝜆_𝑛21𝜆+21_𝑛(1−𝑥)_−𝑎(𝑛∈ℕ∪{0}),(1.17)=^∞_𝑛=0(𝑎)_𝑛𝑛!𝑥^𝑛(𝜆)_𝑛−𝑘,(1.18)=(−1)^𝑘(𝜆)_𝑛(1−𝜆−𝑛)_𝑘(𝑛−𝑘)!=(−1)_𝑘(0≤𝑘≤𝑛;𝑛∈ℕ∪{0}),(1.19)𝑛!(−𝑛)_𝑘(0≤𝑘≤𝑛;𝑛∈ℕ∪{0}).(1.20)

Table 1 

2. Main Transformation Formulae

The generalization of the quadratic transformation (1.6) due to Gauss to be established is (1−𝑥)^2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑏;−4𝑥(1−𝑥)²=2𝑏+𝑗_∞⎤⎥⎥⎥⎥⎥⎦_𝑛=0𝐷_𝑗(𝑎)_𝑛(𝑎−𝑏+(1/2)−[𝑗/2])_𝑛(𝑏+(1/2)+[𝑗/2])_𝑛𝑛!𝑥^2𝑛+2𝑎^∞_𝑛=0𝐸_𝑗(𝑎+1)_𝑛(𝑎−𝑏+(3/2)−[(𝑗+1)/2])_𝑛(𝑏+(1/2)+[(𝑗+1)/2])_𝑛𝑛!𝑥^2𝑛+1for𝑗=0,±1,±2.(2.1)

Also, as usual, [𝑥] represents the greatest integer less than or equal to 𝑥, and its modulus is denoted by |𝑥|. The coefficients 𝐷_𝑗 and 𝐸_𝑗 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=(1−𝑥)^2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑏;−4𝑥(1−𝑥)²2𝑏+𝑗⎤⎥⎥⎥⎥⎥⎦.(2.2) Expressing ₂𝐹₁ as a series and after a little simplification =^∞_𝑘=0(𝑎)_𝑘(𝑏)_𝑘(−1)^𝑘2^2𝑘𝑥^𝑘𝑘!(2𝑏+𝑗)_𝑘(1−𝑥)^{−(2𝑎+2𝑘)}.(2.3) Using Binomial theorem (1.18), we have =^∞_𝑘=0(𝑎)_𝑘(𝑏)_𝑘(−1)^𝑘2^2𝑘𝑥^𝑘𝑘!(2𝑏+𝑗)^𝑘∞_𝑛=0(2𝑎+2𝑘)_𝑛𝑛!𝑥^𝑛,(2.4) which on simplification gives =^∞_∞𝑘=0_𝑛=0(2𝑎+2𝑘)_𝑛(𝑎)_𝑘(𝑏)_𝑘(−1)^𝑘2^2𝑘(2𝑏+𝑗)_𝑘𝑛!𝑘!𝑥^𝑛+𝑘.(2.5) Changing 𝑛 to 𝑛−𝑘 and using the result [1, page 57, lemma 11] ^∞_∞𝑛=0_𝑘=0𝐴(𝑘,𝑛)=^∞_𝑛𝑛=0_𝑘=0𝐴(𝑘,𝑛−𝑘),(2.6) we have 𝑆=^∞_𝑛𝑛=0_𝑘=0(2𝑎+2𝑘)_𝑛−𝑘(𝑎)_𝑘(𝑏)_𝑘(−1)^𝑘2^2𝑘(2𝑏+𝑗)_𝑘(𝑛−𝑘)!𝑘!𝑥^𝑛=^∞_𝑛𝑛=0_𝑘=0Γ(2𝑎+𝑛+𝑘)Γ(2𝑎+2𝑘)(𝑎)^𝑘(𝑏)_𝑘(−1)^𝑘2^2𝑘(2𝑏+𝑗)_𝑘(𝑛−𝑘)!𝑘!𝑥^𝑛.(2.7) Using (1.20) and after a little algebra =^∞_𝑛=0(2𝑎)_𝑛𝑛!𝑥^𝑛𝑛_𝑘=0(−𝑛)_𝑘(2𝑎+𝑛)_𝑘(𝑏)_𝑘(𝑎+(1/2))_𝑘(2𝑏+𝑗)_𝑘𝑘!.(2.8)
Summing up the inner series, we have 𝑆=^∞_𝑛=0(2𝑎)_𝑛𝑛!𝑥^𝑛3𝐹₂⎡⎢⎢⎢⎢⎣𝑎−+𝑛1,2𝑎+𝑛,𝑏;12,2𝑏+𝑗⎤⎥⎥⎥⎥⎦,(2.9) separating into even and odd powers of 𝑥, we have 𝑆=^∞_𝑛=0(2𝑎)_2𝑛(2𝑛)!𝑥^32𝑛𝐹₂⎡⎢⎢⎢⎢⎣𝑎−+2𝑛1,2𝑎+2𝑛,𝑏;1+2,2𝑏+𝑗_∞⎤⎥⎥⎥⎥⎦_𝑛=0(2𝑎)_2𝑛+1(2𝑛+1)!𝑥^32𝑛+1𝐹₂⎡⎢⎢⎢⎢⎣−𝑎2𝑛+−11,2𝑎+2𝑛+1,𝑏;12,2𝑏+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, ±1, ±2, we get, after summing up the series in terms of generalized hypergeometric function, the following interesting results:(i)For 𝑗=0, (1−𝑥)^2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑏;−4𝑥(1−𝑥)²2𝑏⎤⎥⎥⎥⎥⎥⎦=²𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑎−𝑏+12;𝑥₂𝑏+12⎤⎥⎥⎥⎥⎥⎦.(3.1)(ii)For 𝑗=1, (1−𝑥)^2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑏;−4𝑥(1−𝑥)²=2𝑏+1₂⎤⎥⎥⎥⎥⎥⎦𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑎−𝑏+12;𝑥₂𝑏+12⎤⎥⎥⎥⎥⎥⎦+2𝑎𝑥(2𝑏+1)²𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎+1,𝑎−𝑏+12;𝑥₂𝑏+32⎤⎥⎥⎥⎥⎥⎦.(3.2)(iii)For 𝑗=−1, (1−𝑥)^2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑏;−4𝑥(1−𝑥)²=2𝑏−1₂⎤⎥⎥⎥⎥⎥⎦𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑎−𝑏+32;𝑥₂𝑏−12⎤⎥⎥⎥⎥⎥⎦−2𝑎𝑥(2𝑏−1)²𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎+1,𝑎−𝑏+32;𝑥₂𝑏+12⎤⎥⎥⎥⎥⎥⎦.(3.3)(iv)For 𝑗=2, (1−𝑥)^2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑏;−4𝑥(1−𝑥)²=2𝑏+2₂⎤⎥⎥⎥⎥⎥⎦𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑎−𝑏+12;𝑥₂𝑏+32⎤⎥⎥⎥⎥⎥⎦+2𝑎𝑥(𝑏+1)²𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎+1,𝑎−𝑏+12;𝑥₂𝑏+3+4𝑎(𝑎+1)2⎤⎥⎥⎥⎥⎥⎦(𝑏+1)(2𝑏+3)𝑥²²𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎+2,𝑎−𝑏+12;𝑥₂𝑏+52⎤⎥⎥⎥⎥⎥⎦.(3.4)(v)For 𝑗=−2, (1−𝑥)^2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑏;−4𝑥(1−𝑥)²=2𝑏−2₂⎤⎥⎥⎥⎥⎥⎦𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑎−𝑏+32;𝑥₂𝑏−12⎤⎥⎥⎥⎥⎥⎦−2𝑎𝑥(𝑏−1)²𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎+1,𝑎−𝑏+52;𝑥₂𝑏−1+4𝑎(𝑎+1)2⎤⎥⎥⎥⎥⎥⎦(𝑏−1)(2𝑏−1)𝑥²²𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎+2,𝑎−𝑏+52;𝑥₂𝑏+12⎤⎥⎥⎥⎥⎥⎦.(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, (1−𝑥)^2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,12;−4𝑥(1−𝑥)²1⎤⎥⎥⎥⎥⎥⎦=²𝐹₁⎡⎢⎢⎢⎢⎣𝑎,𝑎;𝑥₂1⎤⎥⎥⎥⎥⎦.(3.6)(2)For 𝑗=1, (1−𝑥)^2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,12;−4𝑥(1−𝑥)²=₂2⎤⎥⎥⎥⎥⎥⎦𝐹₁⎡⎢⎢⎢⎢⎣𝑎,𝑎;𝑥₂1⎤⎥⎥⎥⎥⎦+𝑎𝑥²𝐹₁⎡⎢⎢⎢⎢⎣𝑎+1,𝑎;𝑥₂2⎤⎥⎥⎥⎥⎦.(3.7)(3)For 𝑗=2, (1−𝑥)^2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,12;−4𝑥(1−𝑥)²=₂3⎤⎥⎥⎥⎥⎥⎦𝐹₁⎡⎢⎢⎢⎢⎣𝑎,𝑎;𝑥₂2⎤⎥⎥⎥⎥⎦+4𝑎𝑥3²𝐹₁⎡⎢⎢⎢⎢⎣𝑎+1,𝑎;𝑥₂+2𝑎(𝑎+1)2⎤⎥⎥⎥⎥⎦3𝑥²²𝐹₁⎡⎢⎢⎢⎢⎣𝑎+2,𝑎;𝑥₂3⎤⎥⎥⎥⎥⎦.(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 𝑥 replaced by −𝑥), 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 𝑥 by 𝑥/𝑏 and let 𝑏→∞, 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⎡⎢⎢⎢⎢⎢⎣12𝑎,12𝑎+12;4𝑥(1+𝑥)²𝑏+12⎤⎥⎥⎥⎥⎥⎦=(1+𝑥)^𝑎2𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑎−𝑏+1𝑏+12;𝑥2⎤⎥⎥⎥⎥⎥⎦,(4.1) by employing (3.1).

Proof. In order to prove (4.1), we require the following result due to Kummer [4]: ₂𝐹₁⎡⎢⎢⎢⎢⎣𝑎,𝑏;2𝑥2𝑏1+𝑥⎤⎥⎥⎥⎥⎦=(1+𝑥)^𝑎2𝐹₁⎡⎢⎢⎢⎢⎢⎣12𝑎,21𝑎+21;𝑥₂𝑏+12⎤⎥⎥⎥⎥⎥⎦.(4.2) Equation (4.2) is a well-known quadratic transformation recorded in Erdèlyi 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 𝑥 by 2√𝑥/(1+𝑥), then we have ₂𝐹₁⎡⎢⎢⎢⎢⎢⎢⎣𝑎,𝑏;4√𝑥1+√𝑥²2𝑏⎤⎥⎥⎥⎥⎥⎥⎦=1+√𝑥^2𝑎(1+𝑥)^𝑎2𝐹₁⎡⎢⎢⎢⎢⎢⎣12𝑎,12𝑎+21;4𝑥(1+𝑥)²𝑏+12⎤⎥⎥⎥⎥⎥⎦.(4.3) Transposing the above equation, we have ₂𝐹₁⎡⎢⎢⎢⎢⎢⎣12𝑎,12𝑎+21;4𝑥(1+𝑥)²𝑏+12⎤⎥⎥⎥⎥⎥⎦=(1+𝑥)^𝑎1+√𝑥^22𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎢⎣𝑎,𝑏;4√𝑥1+√𝑥²2𝑏⎤⎥⎥⎥⎥⎥⎥⎦.(4.4) Now, in (3.1) first replacing 𝑥 by −𝑥 and then replacing 𝑥 by √𝑥 and using on the right-hand side of (4.4), we get ₂𝐹₁⎡⎢⎢⎢⎢⎢⎣12𝑎,12𝑎+12;4𝑥(1+𝑥)²𝑏+12⎤⎥⎥⎥⎥⎥⎦=(1+𝑥)^𝑎2𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑎−𝑏+1𝑏+12;𝑥2⎤⎥⎥⎥⎥⎥⎦.(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⎡⎢⎢⎢⎢⎣−𝑚,122;4𝑧(1−𝑧)⎤⎥⎥⎥⎥⎦=(𝑚+1)(1−𝑧)𝑧_22𝑚−1𝐹₁⎡⎢⎢⎢⎢⎣−𝑚,−𝑚;1−𝑧𝑧²+(2𝑧−1)𝑧_22𝑚−12⎤⎥⎥⎥⎥⎦𝐹₁⎡⎢⎢⎢⎢⎣−𝑚,−𝑚;1−𝑧𝑧²1⎤⎥⎥⎥⎥⎦,(4.6) in order to derive a quantitative form of the Ishikawa-tdelstin-ó 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 𝑥 by (𝑧−1)/𝑧, we get after a little simplification the following results: ₂𝐹₁⎡⎢⎢⎢⎢⎣2𝑎𝑏,𝑏;4𝑧(1−𝑧)⎤⎥⎥⎥⎥⎦=𝑧_2−2𝑎𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑎−𝑏+12;1−𝑧𝑧²𝑏+12⎤⎥⎥⎥⎥⎥⎦,(4.7)₂𝐹₁⎡⎢⎢⎢⎢⎣=2𝑏𝑧𝑎+,1_−2𝑎𝑏;4𝑧(1−𝑧)⎤⎥⎥⎥⎥⎦⎧⎪⎪⎨⎪⎪⎩₂𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑎−𝑏+12;1−𝑧𝑧²𝑏+1−2𝑎2⎤⎥⎥⎥⎥⎥⎦2𝑏+11−𝑧𝑧²𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎+1,𝑎−𝑏+12;1−𝑧𝑧²𝑏+32⎤⎥⎥⎥⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭,(4.8)₂𝐹₁⎡⎢⎢⎢⎢⎣=2𝑏𝑧𝑎+,2_−2𝑎𝑏;4𝑧(1−𝑧)⎤⎥⎥⎥⎥⎦⎧⎪⎪⎨⎪⎪⎩₂𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎,𝑎−𝑏+12;1−𝑧𝑧²𝑏+1−2𝑎2⎤⎥⎥⎥⎥⎥⎦2𝑏+11−𝑧𝑧²𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎+1,𝑎−𝑏+12;1−𝑧𝑧²𝑏+3+4𝑎(𝑎+1)2⎤⎥⎥⎥⎥⎥⎦(𝑏+1)(2𝑏+3)1−𝑧𝑧²²𝐹₁⎡⎢⎢⎢⎢⎢⎣𝑎+2,𝑎−𝑏+12;1−𝑧𝑧²𝑏+52⎤⎥⎥⎥⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭.(4.9)

Finally, in (4.7), (4.8), and (4.9) if we take 𝑎=−𝑚 and 𝑏=1/2, we get the following very interesting results: ₂𝐹₁⎡⎢⎢⎢⎢⎣121,−𝑚;4𝑧(1−𝑧)⎤⎥⎥⎥⎥⎦=𝑧_22𝑚𝐹₁⎡⎢⎢⎢⎢⎣−𝑚,−𝑚;1−𝑧𝑧²1⎤⎥⎥⎥⎥⎦,(4.10)₂𝐹₁⎡⎢⎢⎢⎢⎣121,−𝑚;4𝑧(1−𝑧)⎤⎥⎥⎥⎥⎦=𝑧_2𝑚⎧⎪⎪⎨⎪⎪⎩₂𝐹₁⎡⎢⎢⎢⎢⎣−𝑚,−𝑚;1−𝑧𝑧²+𝑚11−𝑧⎤⎥⎥⎥⎥⎦𝑧²𝐹₁⎡⎢⎢⎢⎢⎣−𝑚,−𝑚+1;1−𝑧𝑧²2⎤⎥⎥⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭,(4.11)₂𝐹₁⎡⎢⎢⎢⎢⎣1=22,𝑧−_2𝑚𝑚;4𝑧(1−𝑧)⎤⎥⎥⎥⎥⎦⎧⎪⎪⎨⎪⎪⎩₂𝐹₁⎡⎢⎢⎢⎢⎣−𝑚,−𝑚;1−𝑧𝑧²+4𝑚1⎤⎥⎥⎥⎥⎦31−𝑧𝑧²𝐹₁⎡⎢⎢⎢⎢⎣−𝑚,−𝑚+1;1−𝑧𝑧²−2𝑚(1−𝑚)2⎤⎥⎥⎥⎥⎦31−𝑧𝑧²²𝐹₁⎡⎢⎢⎢⎢⎣−𝑚,−𝑚+2;1−𝑧𝑧²3⎤⎥⎥⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭.(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 𝑎𝑏𝑐(𝑐−1)𝑧²𝐹₁⎡⎢⎢⎢⎢⎣𝑎𝑐++11,𝑏+1;𝑧⎤⎥⎥⎥⎥⎦=₂𝐹₁⎡⎢⎢⎢⎢⎣𝑐+𝑎,1𝑏;𝑧⎤⎥⎥⎥⎥⎦−₂𝐹₁⎡⎢⎢⎢⎢⎣𝑎𝑐,𝑏;𝑧⎤⎥⎥⎥⎥⎦(4.13) with 𝑎=𝑏=−𝑚 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.