Some results on confluent hypergeometric functions _mF_m

Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 2, 95 - 101
<strong>Some</strong> Results on Confluent Hypergeometric
Functions mF m
S. Mubeen
National College of Business Administration and Economics
Gulberg-III, Lahore, Pakistan
smjhanda@gmail.com
G. M. Habibullah
National College of Business Administration and Economics
Gulberg-III, Lahore, Pakistan
Abstract
In this paper, we determine an integral representation of generalized
hypergeometric functions m+ 1Fm
by using Legendre’s duplication formula.
Moreover, we use this integral representation to obtain some <strong>results</strong> on
generalized confluent hypergeometric functions mF m.
Mathematic Subject Classification: 33B15, 33C15, 33C20, 33C60
Keywords: Beta and Gamma Functions, Confluent Hypergeometric Functions,
Generalized Hypergeometric Functions, Integral Representations.
1 Introduction
Following Driver and Johnston [1], the authors [2] have also recently determined
the integral representation of generalized confluent hypergeometric functions as
⎛ b b+ 1 b+ m− 1 c c+ 1 c+ m−1
⎞
mFm⎜ , ,....., ; , ,....., ; z⎟
⎝m m m m m m ⎠
1 Γ(
c)
1 m
b−1c−b− zt
=
t ( 1−
t) e dt
Γ( b) Γ( c−b) ∫ ,
0
where Re( c) Re( b) 0, m Z +
> > ∈ .

96 S. Mubeen and G. M. Habibullah
We shall use later the Legendre’s duplication formula [3] to obtain the integral
representation of generalized hypergeometric functions as
m
Π
s=
1
1 ⎛ s −1
⎞
1
2( m−1) −mα
2
Γ ⎜α+ ⎟=
( 2π)
m Γ(
mα)
. (1.1)
⎝ m ⎠
2 An Integral Representation of Generalized Hypergeometric
Functions
We begin with the proof of integral representation for 3F 2.
Re 2c− 2b ⎛ 1 1 ⎞
F ⎜a, b, b+ ; c, c+ ; x⎟
⎝ 2 2 ⎠
> 0 , then
=
Γ
Γ(
2c)
2b Γ 2c−2b 1
2b−1 2c−2b−1 2
−a
∫t
( 1−t) ( 1 −xt
) dt.
(2.1)
Theorem 2.1: If ( )
3 2
( ) ( )
Proof: Suppose that 1
3 2
0
x < . Then
∞ ( a) ( b) ( b+ 1
2 ) k
k k k x
∑
1
k = 0 ( c) ( c+
2 ) k!
k k
1 ( ) ∞
1
( 2) ( ) k ( ) ( 2)
1 ( ) ( )
∑
b b 1
k = 0 ( c k) ( c k )
⎛ 1 1 ⎞
F ⎜a, b, b+ ; c, c+ ; x⎟
=
⎝ 2 2 ⎠
Γ c Γ c+ =
Γ Γ + 2
a Γ b+ k Γ b+ k +
Γ + Γ + + 2
k
x
.
k!
=
Γ
Γ(
2c)
2b Γ 2c−2b ∞ ( a) Γ ( 2b+ 2k) Γ( 2c−2b) k
k
x
∑
, by (1.1)
Γ 2c+ 2 k k!
( ) ( )
Γ(
2 )
( ) ( )
k = 0
( )
∞ ⎛1 2b 2k 1 2c−2b−1 ⎞
+ −
∑ ( a) ⎜ t ( 1 t
k
) dt⎟
⎜∫ ⎟
k = 0 0
=
Γ 2b Γ
c
2c−2 b
⎝
−
x
⎠
k!
1
⎛ 2
k ⎞
∞
Γ( 2c)
( xt
2b−1 2c−2b−1⎜ ) ⎟
= t ( 1−t)
( a) dt
Γ( 2b) Γ( 2c−2 b k
) ∫
⎜∑ 0
k = 0 k!
⎟
⎜ ⎟
⎝ ⎠
k

Confluent hypergeometric functions 97
Γ(
2c)
( 2b) ( 2c 2b)
1
0
2c−2b−1 ( 1 ) ( 1 )
2b−1 2
= t −t −xt
dt
Γ Γ − ∫ .
Theorem 2.2: If ( c b a)
3 2
Re 2 −2− > 0 , then
⎛ 1 1 ⎞
F ⎜a, b, b+ ; c, c+
;1⎟
⎝ 2 2 ⎠
Γ( 2c) Γ( 2c−2b−a) = 2F1( a,2 b;2 c−a; −1)
. (2.2)
Γ 2c−a Γ 2c−2b ( ) ( )
Proof: Equation (2.1) yields that
⎛ 1 1 ⎞
3F2⎜a, b, b+ ; c, c+
;1⎟
⎝ 2 2 ⎠
1
Γ(
2c)
2b−1 2c−2b−1 2
−a
= t ( 1−t) ( 1−t
) dt
Γ 2b Γ 2c−2b ∫
( ) ( )
Γ(
2c)
( 2b) ( 2c 2b)
0
1
2 2 1
( 1 ) ( 1 )
2b−1 c− b−a− −a
= t − t + t dt
Γ Γ − ∫
0
1
Γ( 2c)
2b−1( 1
Γ( 2b) Γ( 2c−2b) ∫
0
∞
2c−2b−a−1 ⎛−a⎞ j1
) ∑ j j ⎝ 1 ⎠
1
Γ( 2c)
∞ 1
⎛−a⎞ 2b+ j −1
2c−2b−a−1 = t −t ⎜ ⎟t
dt
∑ ∫
( 1 )
= ⎜
( 2 ) ( 2 2 ) j
⎟ −
Γ b Γ c− b j 1
1=
0⎝ ⎠0
∞
Γ( 2c) ⎛−a⎞Γ 2b+ j1Γ 2c−2b−a =
( 2b) ( 2c 2b) ∑ ⎜
j
⎟
Γ Γ − ⎝ 1 ⎠ Γ( 2c−
a+ j1)
j = 0
( c) ( c b a)
( ) ( )
( 2 ) ( 2 2 )
( 2c a) ( 2c 2b)
1
1 t t dt
( ) ( )
( a) ( 2b)
( )
j = 0
j
1 1
( ) 1 j
Γ 2 Γ 2 −2 − j 1
1 j − 1
= ∑
Γ 2c−a Γ 2c−2b 2 c−a j !
Γ c Γ c− b−a = 2F1 a b c−a −
Γ − Γ −
Corollary 2.3: If ( )
∞
1
( ,2 ;2 ; 1)
Re 2c 2b n 0, n Z ,
+
− + > ∈ then
⎛ 1 1 ⎞ ( 2c−2b) n
F⎜− nbb , , + ; cc , + ;1 ⎟=
⎝ 2 2 ⎠ 2c
F − n,2 b;2 c+ n;
−1
( )
3 2 2 1
n
.
( )
. (2.3)

98 S. Mubeen and G. M. Habibullah
Theorem 2.4: If ( c b a)
4 3
Re 3 −3− > 0 , then
⎛ 1 2 1 2 ⎞
F ⎜a, b, b+ , b+ ; c, c+ , c+
;1⎟
⎝ 3 3 3 3 ⎠
( 3c) ( 3c ∞
( 3c 3b
a)
−a
( 3b)
j1
) ( 3 3 ) ∑
a c b j j 1 ( )
1= 0⎝
⎠ 3c
a
j1
× F ( − j b+ j c− a+ j − )
Γ Γ − − ⎛ ⎞
= ⎜ ⎟
Γ − Γ − −
Corollary 2.5: If ( c b n)
4 3
,3 ;3 ; 1 .
2 1 1 1 1
Re 3 − 3 + > 0 , then
. (2.4)
⎛ 1 2 1 2 ⎞
F ⎜− n, b, b+ , b+ ; c, c+ , c+
;1⎟
⎝ 3 3 3 3 ⎠
( 3c−3b n ) ( 3 )
n ⎛ n ⎞ b
j1
=
( ) ∑ ⎜
3 j
⎟
c ⎝ 1 ⎠(
3c+
n)
2F1 − j1,3 b+ j1;3 c+ n+ j1;
−1
n j = 0
j
1 1
( )
Similar technique leads to the following generalization of 2.1
Theorem 2.6: If Re( mc − mb)
> 0 , then
⎛ 1 m−1 1 m−1
⎞
m+ 1Fm⎜a,
b, b+ ,..., b+ ; c, c+ ,..., c+ ; x⎟
⎝ m m m m ⎠
1
Γ(
mc)
mb−1 mc−mb−1 m
−a
= t ( 1−t) ( 1 −xt
) dt.
Γ mb Γ mc −mb ∫
( ) ( )
Theorem 2.7: If ( mc mb a)
m+ 1 m
Re − − > 0 , then
⎛ 1 m−1 1 m−1
⎞
F ⎜abb , , + ,..., b+ ; cc , + ,..., c+
;1
m m m m
⎟
⎝ ⎠
Γ( mc) Γ( mc −mb −a)
=
Γ mc −aΓ mc −mb
( ) ( )
0
r−3
⎧ ⎛ ⎞
∞
j
mb j
3
s
a ( mb
m r−
⎜ +
)
j
∑
⎫
⎟
⎛−⎞ j ⎪ ⎛ 1 r−3
⎞ ⎝ s=
1 ⎠j
⎪ r−2
× ∑⎜ ( mc a)
r 3
j
⎟ − ∏ ⎨ ∑ ⎜ −
j
j 1 1
j
⎟⎛
⎞ ⎬
2
1= 0⎝ ⎠ r=
4 ⎪jr2 0 r− mc− a+ j
− = ⎝ ⎠⎜
∑ s ⎟ ⎪
⎩ ⎝ s= 1 ⎠jr−2⎭
⎛ m−2 m−2
⎞
× 2F1⎜− jm−2, mb+ jk; mc− a+ jk;
−1
.
⎜ ⎟
⎝ k= 1 k=
1 ⎠
Re mc mb n 0, n Z ,
+
− + > ∈ , then
. (2.5)
(2.6)
∑ ∑ (2.7)
Corollary 2.8: If ( )

Confluent hypergeometric functions 99
F
m+ 1 m
⎛ 1 m−1 1 m−1
⎞
⎜− nbb , , + ,..., b+ ; cc , + ,..., c+
;1
m m m m
⎟
⎝ ⎠
( mc − mb)
n
=
mc
( )
n
r−3
⎧ ⎛ ⎞
n
j
mb j
3
s
n ( mb
m r−
⎜ +
)
j
∑
⎫
⎟
⎛ ⎞ j ⎪ ⎛ 1
1 r−3
⎞ ⎝ s=
⎠j
⎪ r−2
× ∑⎜ ( )
r 3
j
⎟ mc−a ∏⎨
∑ ⎜
j
j 1 1
j
⎟⎛
− ⎞ ⎬
2
1= 0⎝ ⎠ r=
4 ⎪jr2 0 r mc n j
− = ⎝ − ⎠⎜
+ + ∑ s ⎟ ⎪
⎩ ⎝ s= 1 ⎠jr−2⎭
⎛ m−2 m−2
⎞
× 2F1⎜− jm−2, mb+ jk; mc+ n+ jk;
−1
. (2.8)
⎜ ∑ ∑ ⎟
⎝ k= 1 k=
1 ⎠
3 Consequential Results on Confluent Hypergeometric
Functions
Result 3.1:
1 1
⎛ ⎞
−x e 2F2 ⎜b, b+ ; c, c+ ,; x
⎜
⎟
2 2 ⎟
⎝ ⎠
1
⎛ ∞ ∞ ( ) ( ) ( ) k
−x
⎞⎛
b b+
k 2 k x
⎞
= ⎜ ⎜ ⎟
⎜∑ ⎟
1
n= 0 n! ⎟ ∑
⎝ ⎠⎜k= 0( c) ( c+
2 ) k!
⎟
⎝ k k ⎠
=
∞
∑∑
−
( − )
( )
( b) ( b+
1
2 )
1 ( c) ( c )
n n k
k
x k k x
n− k ! + k!
n= 0k= 0 k 2 k
∞
3 2
n=
0
( −x)
⎛ 1 1 ⎞
= ∑ F⎜− nbb , , + ; cc , + ;1⎟
⎝ 2 2 ⎠ n!
( 2 2 )
( )
n= 0 n
It thus follows that
( )
( ,2 ;2 ; 1)
∞ c−b n
−x
= ∑
2F1 − n b c+ n −
2 c n!
1 1
( 2 2 )
( )
∞
x n
2 2 ⎜
⎟
2 1
n= 0 n
n
n
.
( )
( )
⎛ ⎞ c−b −x
F b, b+ ; c, c+ ,; x = e F − n,2 b;2 c+ n;
−1
.
⎝ 2 2 ⎠ 2 c n!
∑
n

100 S. Mubeen and G. M. Habibullah
Result 3.2:
−x ⎛ 1 2 1 2 ⎞
e 3F3⎜b, b+ , b+ ; c, c+ , c+ ; x⎟
⎝ 3 3 3 3 ⎠
⎛ ∞ n ∞
1 2
( ) ( ) ( ) ( ) k
−x
⎞ ⎛ b b+ b+
k 3 k 3 k x
⎞
= ⎜ ⎟⎜
⎟
⎜∑ 1 2
n 0 n! ⎟⎜∑ = k= 0( c) ( c 3) ( c 3)
k!
⎟
⎝ ⎠⎝
+ +
k k k ⎠
∞ n ⎛( − n) ( b) ( b+ 1) ( b+ 2
3 3)
1
⎞ n
k k k k ( −x)
= ∑∑ ⎜ ⎟
1 2
n= 0k= 0⎜ ( c) ( c+ 3) ( c+
3)
k! ⎟ n!
⎝ k k k ⎠
∞
⎛ 1 2 1 2 ⎞ −x
= ∑ 4F3⎜− n, b, b+ , b+ ; c, c+ , c+
;1⎟
n=
0 ⎝ 3 3 3 3 ⎠ n!
∞ ( 3c−3b n ) ( 3 )
n ⎛ n ⎞ b
j1
= ∑ ( 3c) ∑⎜
j
⎟
⎝ 1 ⎠(
3c+
n)
Hence, it follows that
3 3
n= 0 n j = 0
j
1 1
( −x)
( )
( )
× 2F1 − j1,3 b+ j1;3 c+ n+ j1;
−1
, by 2.6.
n!
⎛ 1 2 1 2 ⎞
F ⎜b, b+ , b+ ; c, c+ , c+ ; x⎟
⎝ 3 3 3 3 ⎠
∞ ( 3 3 n
x c−b) n ⎛ n ⎞
= e ∑ ( 3c) ∑⎜
j
⎟
⎝ ⎠ 3c
n
We Generalize 3.1 and 3.2 as
( 3b)
j1
( + )
n= 0 n j = 0 1
j
1 1
( −x)
( )
× 2F1 − j1,3 b+ j1;3 c+ n+ j1;
−1
.
n!
Result 3.3: For m > 3 , we have
−x ⎛ 1 m−1 1 m−1
⎞
e mFm⎜b, b+ ,..., b+ ; c, c+ ,..., c+ ; x⎟
⎝ m m m m ⎠
( ) ( b) ( b 1 ) ... ( b m−1 ⎛ ∞ ∞
) k
−x
⎞⎛
+ +
k m k m k x
⎞
= ⎜ ⎜ ⎟
⎜∑ ⎟
1
1
0 ! 0(
) ( ) ... m
n n ⎟ ∑
k c c ( c −
⎝ = ⎠⎜ = + + ) k!
⎟
⎝ k m k m k ⎠
n
n
n

Confluent hypergeometric functions 101
=
∞
∑∑
−
( − )
( )
( b) ( b+ 1 ) ... ( b+
m−1
m m )
1
1
( c) ( c ) ... m ( c − )
n n k
k
x k k k x
n− k ! + + k!
n= 0k= 0
k m k m k
∞
m+ 1 m
n=
0
( −x)
⎛ 1 m−1 1 m−1
⎞
= ∑ F ⎜− n, b, b+ ,..., b+ ; c, c+ ,..., c+
;1⎟
⎝ m m m m ⎠ n!
So that
⎛ 1 m−1 1 m−1
⎞
F ⎜b, b+ ,..., b+ ; c, c+ ,..., c+ ; x⎟
⎝ m m m m ⎠
∞
n
x ( mc − mb) ( mb)
n ⎛ n ⎞ j1
= e ∑ ( ) ∑⎜
mc j
⎟(
mc−a ) j ⎝ 1 ⎠
1
m m
References
m
∏
n= 0 n j = 0
1
r−3
⎧ ⎛ ⎞
j
mb j
r−3
⎜ + s
j
∑
⎫
⎟
⎪ ⎛ r−3
⎞ ⎝ s= 1 ⎠j⎪
r−2
∑ ⎜ r−3
j
⎟⎛
⎞
⎪j2 r 2 0 r− mc+ n+ j
− = ⎝ ⎠ ∑ s ⎪
⎝ s= 1 ⎠jr−2
× ⎨ ⎬
r=
4
⎜ ⎟
⎩ ⎭
( x)
⎛ m−2 m−2
⎞ −
2F1 ⎜
jm−2 mb ∑ jk mc n ∑ jk
⎟
k= 1 k=
1
× − , + ; + + ; −1
.
⎝ ⎠ n!
[1] K. A. Driver and S. J. Johnston, An integral representation of some
hypergeometric functions, Electronic Transactions on Numerical Analysis,
25(2006), 115-120.
[2] G. M. Habibullah and S. Mubeen, p
L − bounded operators involving
generalized hypergeometric functions, Integral Transforms and Special
Functions, Vol. 22(2), 143-149.
[3] E. D. Rainville, Special Functions, The Macmillan Company, New York,
1960.
Received: June, 2011
n
n
.