American Journal of Computational and Applied M athematics 2013, 3(3): 182-185
DOI: 10.5923/j.ajcam.20130303.05
A Note on Hypergeometric Functions of One and Two
Variables
Pankaj Srivastava, Priya Gupta*
Department Name of M athematics, M otilal Nehru National Institute of Technology, Allahabad, 211004, India
Abstract The present paper deals with transformations of hypergeometric function of one variable and Kampe de Feriet
hypergeometric function of t wo variab les by using transformations between two terminating Saalschutzian 4 F3 (1) series.
Keywords Hypergeometric Functions, Transformation Formu lae, Kampe de Feriet Hypergeo metric Function
1,
if n = 0,

(a) n = 
a (a + 1)(a + 2)...(a + n − 1), if n ∈ N
1. Introduction
Hypergeometric function is a beautiful tool of special
function that plays an important role in the field o f analysis.
The transformation theory played a major ro le to provide a
platform for the development of beautiful transformat ion. It
is impo rtant to mentation that whenever generalized
hypergeometric function reduces to a gamma function, the
results are very important fro m application point of view in
mathematics, statistics and mathematical physics. On
account of usefulness, hypergeometric function have
already explo red to the same extent by a number of special
function experts notably C. F. G auss, M. M. Kummer, S.
Ramanujan, B. C. Berndt[1], G. N. Watson[5], W. N.
Bailey[18], E. D. Rainville[2], L. J. Slater[9], H. Exton[6],
G. Gasper[4], M . Rah man, G. E. Andrews[3], R. P.
Agrawl[13], R. Y. Denis et al.[15], S. N. Singh[16], H. M.
Srivastava et al.[7], R. P. Singh[14], M. A. Rakha et al.[10],
Yong S. Kim et al.[19], Pankaj Srivastava[11], R. K.
Saxena et al.[12], S. P. Singh[17], Kung-Yu Chen et al.[8]
etc. In the present paper transformat ions of hypergeometric
function of one and two variab les established this paper
should be useful fro m the applicat ion point of view.
2. Definitions and Notations
A generalized hypergeometric function is defined as
 a1 a2 ...a p  ∞ (a1 ) n (a2 ) n ...(a p ) n x n
F
; x = ∑
p q 
 b1 b2 ...bq  n =0 (b1 ) n (b2 ) n ...(bq ) n n !
Where
* Corresponding author:
[email protected] (Priya Gupta)
Published online at http://journal.sapub.org/ajcam
Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved
The series converges when none of denominator
parameter are zero or negative integer. It is converges for all
x when p<q, for |x|<1, it is converges when p=q+1and for
|x|<1 it is converges when Re(∑b i -∑ai )>0 and it is only
converges for x=0 when x=0 when p>q+1 unless it reduces
to a polynomial.
A Kampe de Feriet hypergeometric function of two
variables is defined as,
(a p ) (bq ) (bk ')

; x, y 
Fl:pm:q;n;k 
 (cl ) (d m ) (d n ')

(a p ) (bq ) (bk ')

=F
; x, y 
 (cl ) (d m ) (d n ')

r s
∞ [( a )] [(b )] [(b ')]
p r+s
q r
k s x y
= ∑
r , s =0 [(cl )]r + s [( d m )]r [( d n ')]s r ! s !
Where (ap ) stands for the sequence of p parameters a1 ,…ap
and |x|<1, |y|<1, l+m+n <p+q+k+1 for convergence.
We shall use the fo llo wing known t ransformat ions and
identity in our analysis as given in[1, 2, 4, 9]
 a, b, −n  (c − a)n (d − a)n
;1 =
3 F2 
(c ) n ( d ) n
(1)
c d

a,
−n 
1 − c − d + a + b − n,
;1

1+ a − d − n
 1 + _ a − c − n,

([9], 2.5.11).
3 F2
 a + c,
4 F3 
 a + b + c,
=
b + c,
d
−n 
;1
e, 1 + c + d − e − n

b,
d,
−n 
 a,
(e − d ) n (e − c ) n
F
;1
(e)n (e − d − c)n 4 3  a + b + c, e − c, 1 + d − e − n

([9], 2.4.1.7).
(2)
American Journal of Computational and Applied M athematics 2013, 3(3): 182-185
 2a,

2b,
c,
4 F3  2c a + b + 1 a + b + 1 − c − n
2
2

−n

;1 =

183
 b,
1 −c−n
c − b,
−n 
(a + 12)n (c − a + 12)n
2
;1 (3)
F3 
4
1 1 − a − n, 1 + a − c − n
(a + b + 12)n (c − a − b + 12)n
c + 2

2
2
([9], 2.5.22).

2a,
4 F3  a + b + 1

2

2b,
c + 12
−n  (c − a)n (c − b)n
a,
b,
;1 =
F3 
4
2c 1 + a + b − c − n
 (c)n (c − a − b)n
 a + b + 12 1 + a − c − n,
([9], 2.5.25).
3 F2
([4], 3.1.1).
4 F3
([1], Ch.10, eq. 6.3).
1
2 + a + b − 2c − n
1+ b − c − n
−n 
;1 (4)
b,
−n 
 a, b, −n  (d − b)n
 c − a,
;1 =
;1

3 F2 
1+ b − d − n
(d )n
 c, d

 c,

(5)
d − b,
c, − n 
 a, b, c, − n  (e − c) n ( f − c) n
 d − a,
;1 =
;1
4 F3 
 d , e, f
(e ) n ( f ) n


1 − e + c − n, 1 − f + c − n, d

n
∞
∑=
∑ A(k , n)
=
n 0=
k 0
([2], Ch.4, lemma 10(2)).
∞

(6)
∞
∑ ∑ A(k , n + k )
(7)
=
n 0m
= 0
3. Main Result
In this section we shall establish following results.
Theorem 3.1:
∞
∞ (c − a) (d − a) z n (a) z m (c + d − a − b)
z n (a) (b) (− z )m
n + m (n + m)! Ω
=
n+ m
∑ n! (cm)m (dm)m m! (n + m)!Ωn+m ∑ (c + dn− a − b)nnn! mm!
c
d
(
)
(
n + m )n + m
n, m 0 =
n, m 0
 a, b
e z 2 F2 

c + d − a − b
; −z =
F
:
c − a,
d −a ; a
d : c + d −a −b
; −
c,
 c, d


d −a 
c,
d : − ; a, b
 c − a,

(1 − z )− a 2 F1 
; z F 
=
: − ; c, d
c
d
a
b
c
d
a
b
+
−
−
+
−
−



Theorem 3.2:

(i)
; z, z 
(ii)

; z, − z 

(iii)

(e − d )n z n (a)m (b)m (d )m z m
(n + m)!
Ω
n!
( a + b + c ) m (e − c ) m m ! (e − d ) n + m n + m
n , m =0
∞
∑
(e − c − d )n z n (a + c)m (b + c)m (d )m z m (e)n+ m (n + m)!
= ∑
×
Ω
n!
( a + b + c ) m (e ) m m !
(e − c ) m (e − d ) m n + m
n , m =0
∞
a,
b, d 
a + c, b + c, d
 e : e−c−d ;

; z =
; z, z 
F
,
:
;
,
a
b
c
e
c
e
c
a
b
c
e
+
+
−
−
+
+





(1 − z )d −e 3 F2 
 a + c,
b + c, d 
a,
b, d
e − c : e − d ;

; z =
; z, z 
F
: − ; a + b + c, e − c
e
 a + b + c,

 e

(1 − z )c+ d −e 3 F2 
Theorem 3.3:
(i)
(ii)
(iii)
(c − a − b + 12)n z n (2a)m (2b) (c)m z m
(n + m)!
Ω
1
(a + b + 2)m (2c)m m! (c − a − b + 12)n + m n + m
n!
n, m =0
∞
∑
=
(a + 12)n (c − a + 12)n z n (c − b)m (b)m z m
∑
(c + 12)n n !
(c + 12)m m!
n, m =0
∞
(c + 12)n + m (n + m)!
Ω
(a + b + 12)n + m (c − a − b + 12)n + m n + m
(i)
Pankaj Srivastava et al.: A Note on Hypergeometric Functions of One and Two Variables
184
2b,
c 
 c + 12

: a + 12 c − a + 12 ;
b,
c −b
; z =
F
;
z
,
z

1
1
1
; c + 12
 2c, a + b + 2

a + b + 2 : c + 2

(ii)
c −b 
2b,
c
a + 12 , c − a + 12 
 a + b + 1 2 : c − a − b + 1 2 ; 2a,

; z  2 F1 
;
z
=
F
; z, z 


1
1
1
:
;
2
,
c
+
c
+
−
c
a
+
b
+
2
2
2





(iii)
(1 − z )a +b−c−
 b,
2 F1 
c +
1
2
1
2
 2a,
3 F2 
Theorem 3.4:
m
(c − a − b)n z n (2a)m (2b) (c + 12)m z
(n + m)!
Ω n+ m
1 ) (2c) m! (c − a − b)
n
!
(
a
+
b
+
n+ m
m
2 m
n,m=0
∞
(n + m)!
(c − a) (c − b) z n (a) (b) z m (2c − a − b + 1 )
∑ (2c − an− b + 1 n)n n! (a + bm+ 1 m)m m! (c)n+m (c −2an−+bm)n+m Ωn+m
2
2
n,m=0
(i)
 2c − a − b + 1 :

2a,
2b, c + 1 2 
c − a,
c −b ;
a,
b
2
; z =
F
; z, z 
1
c
: 2c − a − b + 1 2
; a + b + 12
 a + b + 2 , 2c



(ii)
∞
∑

(1 − z )a+b−c 3 F2 
c − a,

2 F1  2c − a − b + 1

2

c −b 
; z  2 F1 


a,
b 
c
:
2a
2b c + 12 ; c − a − b
; z = F 
; z, z 
1
1
1
a
+
b
+
2
c
−
a
−
b
+
:
a
+
b
+
2
c
,
;
−

2
2
2




(iii)
Theorem 3.5:
∞
∞
(d − b)n z n (c − a)m (b)m z m
z n (a)m (b)m (− z )m
(n + m)!
(
n
m
)!
=
+
Ω
Ω
∑ n ! (c ) m ( d ) m m !
∑
n+ m
n!
m!
(c ) n + m ( d ) n + m n + m
n,m 0 =
n,m 0
 a, b

 − : c − a, b ; d − b

F
; −z =
; z, z 
−
c
;
 c, d

d :

(ii)
 c − a, b 
 d : a, b ; −

(1 − z )b−d 2 F1 
; z = F 
; z, − z 
 c

 − : c, d ; −

(iii)
e z 2 F2 
Theorem 3.6:
∞
∑
n,m
(i)
∞
( e − c ) n ( f − c ) n z n ( d − a ) m ( d − b ) m (c ) m ( − z ) m
z n (a)m (b)m (c)m (− z )m
(n + m)!Ωn+ m =
∑
n ! ( d ) m (e) m ( f ) m m !
n!
(d )m m !
0=
n,m 0
(i)
(n + m)!
Ω
(e) n + m ( f ) n + m n + m
 a, b, c 
−

: e − c, f − c ; d − a, d − b, c
=
e z 3 F3 
; z F 
; z, − z 
;
d
 d , e, f 
e, f :

2 F0
 e − c,

 −
f −c 
e,
 d − a, d − b, c

;=
z  3 F1 
; −z F 

 d

−
f
: − ; a, b, c

; z, − z 
: − ; d , e, f

(ii)
(iii)
As an illustration we give p roof of theorem (3.1).
Let Ωn is an arb itrary sequence of comp lex nu mbers, mu ltip lying both sides of (1) by zn Ωn and summing over n fro m 0 to ∞
we get,
n (a) (b) (−n)
m
m
m z nΩ
n
(
c
)
(
d
)
m
!
m
m
m 0
=n 0 =
∞
∑ ∑
∞ (c − a ) ( d − a )
n (1 − c − d + a + b − n) (a) (−n)
n
n
m
m
m z nΩ
n
(
c
)
(
d
)
(1
+
a
−
c
−
n
)
(1
+
a
−
d
−
n
)
m
n
n
m
m !
m 0
=n 0
=
∑
∑
Making use of the identity (7) and (a-n )n=(-1)n(1-a)n in above we get after so me simp lificat ion,
∞
∞
(c − a) (d − a) z n (a) z m (c + d − a − b)n+ m (n + m)!
z n (a) (b) (− z )m
=
Ωn+ m
∑ n! (cm)m (dm)m m! (n + m)!Ωn+m ∑ (c + d n− a − b)nnn! mm!
(c ) n + m ( d ) n + m
n ,m 0 =
n ,m 0
Equation (I) holds provided that both sides do exist. In particu lar, choosing
get,
 a, b
e z 2 F2 
 c, d

c + d − a − b


; −z =
F
c,
Ωn =
(i)
1
and after so me simp lification we
n!
:
c − a,
d −a ; a
d : c + d −a −b
; −

; z, z 

(ii)
American Journal of Computational and Applied M athematics 2013, 3(3): 182-185
While choosing
Ωn =
(c ) n ( d ) n
in eq. (I)
(c + d − a − b) n n!
and after some simp lificat ion we get,
d −a 
 c − a,
(1 − z ) − a 2 F1 
; z
c + d − a − b

(iii)
c,
d : − ; a, b


F
; z, − z 
: − ; c, d
c + d − a − b

Similarly one can establish theorems (3.2), (3.3), (3.4),
(3.5) and (3.6) by suitable transformation fro m (2) to (6) and
appropriate selection of Ωn .
4. Conclusions
The transformations of hypergeometric function of one
and two variables established in this paper should be useful
fro m the application point of view.
185
Ramanujan’s integral formula, Adv. Stud. Contemp. M ath. 18
(2009), 113-125.
[8]
Kung-Yu Chen, Shuoh-Jung Liu, H. M . Srivastava, Some
double-series identities and associated generating-function
relationships, Applied M athematics Letters, 19 (2006)
887-893.
[9]
L. J. Slater, Generalized Hypergeometric Functions,
Cambridge University Press, Cam- bridge, London and New
York, 1966.
[10] M . A. Rakha, Adel K. Ibrahim, and Arjun K. Rathie, On the
computations of contiguous relations for 2F1 hypergeometric
series, Commun. Korean M ath. Soc. 24 (2009), No.2,
pp.291-302.
[11] Pankaj Srivastava, Certain transformations of generalized
Kampe de Feriet function Vijnana Parishad Anusandan
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[12] R. K. Saxena, C. Ram and N. Dudi, Some results associated
with a generalized hyper- geometric function, Bulletin of Pure
and Applied Sciences. Vol.24 (No.2) 2005, 305-316.
[13] R. P. Agarwal, Special function and their application,
Keynote address, Proceedings of National Symposium on
special Function and their application (Gorakhpur 1986).
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