ARJUN, K. RATHIE

MEDHAT, A. RAKHA

NEETHU, P

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ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 3 (2014), no.1, 17
ISSN 1857-8365
ON TWO FORMULAS CONTIGUOUS TO A TRANSFORMATION
DUE TO BAILEY
NEETHU. P, MEDHAT A. RAKHA1 AND ARJUN K. RATHIE
Abstract. The aim of this short research note is to provide two interesting formulas
contiguous to the following transformation due to Bailey (written here in corrected
form) viz.
(1  x) 1 2F1
"
1; b;
 
4x
(1 x)2
2b+ 1
2
;
#
= 2F1
"
1; 3
4
  b;
x2
3
4
+ b;
#
 
x(4b  1)
(4b+ 1)
2F1
"
1; 5
4
  b;
x2
5
4
+ b;
#
The results are derived with the help of two formulas contiguous to classical Whip-
ple's summation theorem on the sum of a 3F2 obtained earlier by Lavoie, et al.
The results established in this short research note are simple, interesting, easily
established and may are useful.
1. Introduction
In the theory of hypergeometric and generalized hypergeometric series, classical sum-
mation theorems such as those of Gauss, Gauss second, Kummer and Bailey for the series
2F1; Watson, Dixon, Whipple and Saalschütz for the series 3F2 and others play an im-
portant role. Several formulae were given by Gauss and Kummer expressing the product
of two hypergeometric series, such as e x 1F1(x) as a series of the type 1F1( x) and
(1 + x) p 2F1

4x
(1 + x)2

as a series of the type 2F1(x). By employing the above mentioned classical summation
theorems, in 1928, Bailey [1] made a systematic search for several formulae. Evidently
if the product of two hypergeometric series can be expressed as a hypergeometric series
with argument x, the coecient of xn in the product must be expressible in terms of the
Gamma function.
1corresponding author
2010 Mathematics Subject Classication. 33C05, 33C20, 33C70.
Key words and phrases. Generalized hypergeometric functions, Bailey's transformation, Whipple's
summation theorem.
1
2 NEETHU. P, M. A. RAKHA AND A. K. RATHIE
Recently good progress has been done in the direction of generalizing and extending
the above mentioned classical summation theorems. For this, we refer to the research
papers [2] - [5].
In our present investigation, we are interested in the following transformation due to
Bailey [1]
(1  x) 1 2F1
2
64 1; b;   4x(1 x)2
2b+ 12 ;
3
75
= 2F1
2
4 1; 34   b; x2
3
4 + b;
3
5  x(4b  1)
(4b+ 1)
2F1
2
4 1; 54   b; x2
5
4 + b;
3
5 :(1.1)
Bailey [1] obtained this result by employing the following classical Whipple's summation
theorem [8] viz
3F2
2
4 a; b; c; 1
e; f ;
3
5
=
  (e)   (f)
22c 1 
 
1
2a+
1
2e

 
 
1
2a+
1
2f

 
 
1
2b+
1
2e

 
 
1
2b+
1
2f
(1.2)
provided a+ b = 1 and e+ f = 1 + 2c.
In 1996, Lavoie, et al. [5] generalized the above mentioned classical Whipple's sum-
mation theorem (1.2) and have obtained explicit expressions of
(1.3) 3F2
2
4 a; b; c; 1
e; f ;
3
5
provided a+ b = 1 + i+ j and e+ f = 2c+ i+ 1 for i; j = 0;1;2;3.
In our present investigation, we shall require the following results which can be ob-
tained from (1.3) by taking i = 1; j =  1 and i =  1; j = 1 respectively, these are
 For i = 1; j =  1
3F2
2
4 a; b; c; 1
e; f ;
3
5 =   (e)   (f)   (e  c  1)
22a  (e  a)   (f   a)   (e  c)

8<
:
 
 
e
2   a2 + 12

 

f
2   a2

 
 
e
2 +
a
2   12

 

f
2 +
a
2
    
 
e
2   a2

 

f
2   a2 + 12

 
 
e
2 +
a
2

 

f
2 +
a
2   12

9=
;(1.4)
provided a+ b = 1; e+ f = 2c+ 2:
ON TWO FORMULAS CONTIGUOUS TO A TRANSFORMATION. . . 3
 For i =  1; j = 1
3F2
2
4 a; b; c; 1
e; f ;
3
5 =   (e)   (f)   (c  1)
22a  (e  a)   (f   a)   (c)

8<
:
 
 
e
2   a2 + 12

 

f
2   a2

 
 
e
2 +
a
2   12

 

f
2 +
a
2
 +  
 
e
2   a2

 

f
2   a2 + 12

 
 
e
2 +
a
2

 

f
2 +
a
2   12

9=
;(1.5)
provided a+ b = 1; e+ f = 2c:
The aim of this short research note is to nd the explicit expressions of
(1  x) 1 2F1
2
64 1; b;   4x(1 x)2
2b+ 12 + l;
3
75
for l = 1.
The results are derived with the help of contiguous Whipple's summation theorems
(1.4) and (1.5). The results established in this short research note are simple interesting,
easily established and may be useful.
2. Main Results
The transformation formulas closely related to the Bailey's transformation to be es-
tablished in this short research note are given by
(1  x) 1 2F1
2
64 1; b;   4x(1 x)2
2b+ 32 ;
3
75
=
1
2(2b+ 1)
3F2
2
4 1; 54 ; 14   b; x2
1
4 ;
5
4 + b;
3
5+ (4b+ 1)
2(2b+ 1)
2F1
2
4 1; 34   b; x2
3
4 + b;
3
5
+
3(4b+ 1)x
2(2b+ 1)(4b+ 3)
3F2
2
4 1; 74 ; 34   b; x2
3
4 ;
7
4 + b;
3
5  (4b  1)x
2(2b+ 1)
2F1
2
4 1; 54   b; x2
5
4 + b;
3
5
(2.1)
4 NEETHU. P, M. A. RAKHA AND A. K. RATHIE
and
(1  x) 1 2F1
2
64 1; b;   4x(1 x)2
2b  12 ;
3
75
=
1
4(1  b) 3F2
2
4 1; 54 ; 54   b; x2
1
4 ; b+
1
4 ;
3
5  (4b+ 3)
4(1  b) 2F1
2
4 1; 74   b; x2
b  14 ;
3
5
+
3(4b  3)x
4(1  b)(4b  1) 3F2
2
4 1; 74 ; 74   b; x2
3
4 ;
3
4 + b;
3
5+ (4b  5)x
4(1  b) 2F1
2
4 1; 94   b; x2
b+ 14 ;
3
5
(2.2)
Proof. In order to derive (2.1), we proceed as follows. Denoting the left-hand side of
(2.1) by S, we have
S = (1  x) 1 2F1
2
64 1; b;   4x(1 x)2
2b+ 32 ;
3
75 :
Expressing 2F1 as a series, we have after some algebra
S =
1X
n=0
(b)n( 1)n22n 
2b+ 32

n
xn(1  x) (2n+1);
using Binomial theorem, we have
S =
1X
n=0
1X
m=0
(b)n( 1)n22n 
2b+ 32

n
(2n+ 1)m
m!
xn+m:
Now, replacing m by m  n and using the known result [6, Lemma 10, p. 56 Eq. (1)]
1X
n=0
1X
k=0
A(k; n) =
1X
n=0
nX
k=0
A(k; n  k);
we have
S =
1X
m=0
mX
n=0
(b)n( 1)n22n 
2b+ 32

n
(2n+ 1)m n
(m  n)! x
m;
using the elementary identities [6]
(a)m n =
( 1)n(a)m
(1  a m)n
;
(m  n)! = ( 1)
nm!
( m)n
and
  (2z) =
22z 1  (z)  
 
z + 12

p

;
we have, after some simplication
S =
1X
m=0
xm
mX
n=0
( m)n(1 +m)n(b)n 
1
2

n
 
2b+ 32

n
n!
:
ON TWO FORMULAS CONTIGUOUS TO A TRANSFORMATION. . . 5
Summing up the inner series, we have
S =
1X
m=0
xm 3F2
2
4  m; 1 +m; b; 1
1
2 ; 2b+
3
2 ;
3
5 :
Now, separating into even and odd powers of x, we have
S =
1X
m=0
x2m 3F2
2
4  2m; 1 + 2m; b; 1
1
2 ; 2b+
3
2 ;
3
5
+ x
1X
m=0
x2m 3F2
2
4  2m  1; 2m+ 2; b; 1
1
2 ; 2b+
3
2 ;
3
5 :
Now assume that
(2.3) S = A+B;
then we have
A =
1X
m=0
x2m 3F2
2
4  2m; 1 + 2m; b; 1
1
2 ; 2b+
3
2 ;
3
5 :
It is now, easy to see that, the 3F2 can be evaluated with the help of the contiguous
Whipple's summation theorem (1.4) by taking a =  2m; b = 1 + 2m; c = b; e = 12 and
f = 2b+ 32 and after some simplication, we get
A =
1
2(2b+ 1)
1X
m=0
(1)m
 
5
4

m
 
1
4   b

m 
1
4

m
 
5
4 + b

m
m!
x2m
+
(4b+ 1)
2(2b+ 1)
1X
m=0
(1)m
 
3
4   b

m 
3
4 + b

m
m!
x2m
summing up the series, we have
A =
1
2(2b+ 1)
3F2
2
4 1; 54 ; 14   b; x2
1
4 ; b+
5
4 ;
3
5
+
(4b+ 1)
2(2b+ 1)
2F1
2
4 1; 34   b; x2
3
4 + b;
3
5 :(2.4)
In exactly the same manner, we can show that
B =
3(4b+ 1)x
2(2b+ 1)(4b+ 3)
3F2
2
4 1; 74 ; 34   b; x2
3
4 ;
7
4 + b;
3
5
  (4b  1)x
2(2b+ 1)
2F1
2
4 1; 54   b; x2
5
4 + b;
3
5 :(2.5)
6 NEETHU. P, M. A. RAKHA AND A. K. RATHIE
Substituting the results of A and B from (2.4) and (2.5) in (2.3), we easily arrive at the
right-hand side of our rst main result (2.1). This completes the proof of (2.1).
In exactly the same manner, the result (2.2) can be established by employing the
contiguous Whipple's summation theorem (1.5), so we prefer to omit the details. 
Clearly the results (2.1) and (2.2) are closely related to the Bailey's transformation
(1.1).
Authors' Contributions
The authors have equal contributions to each part of this paper. All authors have read
and approved the nal manuscript.
Competing Interests
The authors declare that they have no competing interests.
References
[1] W. N. Bailey: Product of generalized hypergeometric series, Proc. London Math. Soc. (2), 28
(1928), 242  254.
[2] Y. S. Kim, M. A. Rakha, A. K. Rathei: Extensions of classical summation theorems for the
series 2F1;3 F2 and 4F3 with applications in Ramanujan's summations, Int. J. Math. & Math. Sci.,
ID 309503 (2010) , 26 pages.
[3] J. L. Lavoie, F. Grondin, A. K. Rathie: Generalizations of Watson's theorem on the sum of
a 3F2, Indian J. Math., 34 (1992), 23  32.
[4] J. L. Lavoie, F. Grondin, A. K. Rathie, K. Arora: Generalizations of Dixon's theorem on
the sum of a 3F2, Math. Comp. American Math. Society, 62, No. 205 (1994), 267  276.
[5] J. L. Lavoie, F. Grondin, A. K. Rathie: Generalizations of Whipple's theorem on the sum of
a 3F2, J. Comput. Appl. Math., 72 (1996), 293  300.
[6] E. D. Rainville: Special Functions, The Macmillan Company, New York, 1960.
[7] M. A. Rakha, A. K. Rathie: Generalizations of classical summation theorems for the series 2F1
and 3F2 with applications, Integral Transform and Special Functions, 22 (11) (2011) 823  840.
[8] L. J. Slater: Generalized hypergeometric functions, Cambridge University Press, Cambridge,
1966.
Department of Mathematics
School of Mathematical and Physical Sciences
Central University of Kerala, Riverside Transit Campus
Padennakkad P.O. Nileshwar, Kasaragod - 671 328, Kerala - INDIA
E-mail address: neethukanayi82@gmail.com
Department of Mathematics and Statistics
College of Science
Sultan Qaboos University
P.O.Box 36 - Al-Khoud 123
Muscat - Sultanate of Oman
E-mail address: medhat@squ.edu.om
ON TWO FORMULAS CONTIGUOUS TO A TRANSFORMATION. . . 7
Department of Mathematics
School of Mathematical and Physical Sciences
Central University of Kerala, Riverside Transit Campus
Padennakkad P.O. Nileshwar, Kasaragod - 671 328, Kerala - INDIA
E-mail address: akrathie@gmail.com


ON TWO FORMULAS CONTIGUOUS TO A TRANSFORMATION DUE TO BAILEY

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ON TWO FORMULAS CONTIGUOUS TO A TRANSFORMATION DUE TO BAILEY

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