STUDYOF A GENERATING FUNCTION
INVOLVING GENERALISED
LAURICELLA FUNCTION
Ekta Mittal
1, Sunil Joshi
2,Rupakshi Mishra Pandey
31Department of Mathematics, The IIS University, Jaipur, Rajasthan,India 2Department of Mathematics & Statistics, Manipal University Jaipur, Rajasthan ,India
3
Department of Applied Mathematics, Amity University Uttar Pradesh, Noida, India
ekta.jaipur@gmail.com, sunil.joshi@manipal.edu, rmpandey@amity.edu
Abstract.Recently, Generating function of the extended second Appllehypergeometricfunction was introduced by S. D.Purohit, R.K.Parmar and K.S. Nishar, here in the present investigations we established some generating functions involving generalised extendedLauricellafunction
, ,
1 2,...,
; ,
1 2,...,
; ,
1 2,...,
; ;
A n n n
F a b b
b c c
c x x
x p
. Further we develop some certain interesting special casesKey words: Extended Beta function, extendedhypergeometric function, Lauricellafunction.
1. INTRODUCTION
Some extensions of the well-known special functions have been considered by several authors (See [1–6]). In 1994, Chaudhry and Zubair [1] presented the following extension of gamma function.
(1)
In 1997, Chaudhry et al.[2] presented the following extension of Euler’s Beta function.
1 1
1
0
R 0; R 0; R 0
, ;
1
xp
,
.
(1
)
y x
p x y
p
B x y p
t
t
e
dt
t
t
(2)
In 2004, Chaudhry et al. [3] used B x y p( , ; )to extend the hypergeometric functions as follows.
0
1
, ; ;
,
;
,
,
!
n
p n
n
z
F a b c z
a
B b n c b p
B b c b
n
z 1; R c R b 0; R p 0 .
(3)
In 2010, Özarslan and Özergin [7] extended the familiar Appellhypergeometric function using
B x y p
( , ; )
as given below.
2
, 0
,
;
'
, '
';
, , '; , '; , ;
,
,
', '
'
! !
m n m n
m n
B b m c b p B b
n c
b p x y
F a b b c c x y p
a
B b c b B b c
b
m n
1
1
0
;
xexp
,
,
(Re
> 0, Re
> 0) .
x p
t
t
t
pt
dt
x
p
In 2011, Lee et al.[9], presented the generalisation of extended beta function (2) and hypergeometric function
(3) as follows: 1
1 1
0
( , ; ; )
(1
)
exp
,
(1
)
x y
p
B x y p
t
t
dt
t
t
R x ,R y 0,R p 0,R 0 .
(5)and
;
0
1
, ; ;
,
; ;
,
,
!
n n
p
n
z
F
a b c z
a
B b n c b p
B b c b
n
z 1,R c R b 0,R p 0,R 0 .
(6)
Recently R. K. Parmar (2014) generalised the extended Appellhypergeometric function by usingB x y p
, , ;
,
2
, 0
,
; ;
'
, '
'; ;
, , '; , '; , ; ;
,
,
', '
'
! !
m n m n
m n
B b m c b p
B b
n c
b p
x
y
F a b b c c x y p
a
B b c b B b c
b
m n
x y 1;R p 0,R 0 .
(7)
Here we also use
B x y p
, , ;
in the firstLauricella function such as
1 1 1
( )
, ,.... ; ,... ; ,...,
n n n; ;
n A
F
a b
b c
c x
x p
1
1
1
1 1 1 1 2 2 2 2
...
,..., 0 1 1 1 2 2 2
1 1
;
; ;
;
; ;
...
;
; ;
;
;
...
;
...
,
!
!
n
n
n
n n n n m m
m m n n n
m m n
n
a
B b
m c
b p
B b
m c
b p
B b
m c
b p
B b c
b B b c
b
B b c
b
x
x
m
m
where
x1 ... xn 1 .
(8)When
1,
equations (5), (6),(7) yields extended Beta function,Gauss’shypergeometricandAppellhypergeometric functions [5,8] respectively.If we put
1,
p
0,
in equation (9), we get the Lauricella function.2. MAIN RESULT
Here we obtain some generating function for the generalised extendedLauricella function
( )
1 1 1
, ,.... ; ,... ; ,... ; ;
n
A n n n
Theorem-1: The following function hold true
1 1 1
0
1 2
1 1
, ,.... ; ,... ; ,... ; ;
!
1
, ,.... ; ,... ;
,
...,
; ;
,
1
1
1
k n
A n n n
k k
n n
A n n
t
F
k b
b c
c x
x p
k
x
x
x
t
F
b
b c
c
p
t
t
t
where{R p
0,R
0,C and t 1}.
Proof:Using definition (8) in the left hand side,we have
1 1 1
0
, ,.... ; ,... ; ,... ; ;
!
k nA n n n
k k
t
F
k b
b c
c x
x p
k
0!
k k kt
k
1 1 11 1 1 1 2 2 2 2
... 1
1 1 1 2 2 2 1
,..., 0 ; ; ; ; ; ; ... ; ; ; .. , , ... , ! ! n n n m m
n n n n
m m n
n n n n
m m
k B b m c b p B b m c b p B b m c b p x x
B b c b B b c b B b c b m m
Changing order of summation
1
1 1 1 1 2 2 2 2
,..., 0 1 1 1 2 2 2
;
; ;
;
; ;
...
;
; ;
,
,
...
,
n
n n n n
m m n n n
B b
m c
b p
B b
m c
b p
B b
m c
b p
B b c
b B b c
b
B b c
b
11 1 ... 0 1
.
...
!
!
!
n nm m k
n
k m m
k n
x
x
t
k
m
m
k
Using the identity
1 2 ... 1 2 ... 1 2
...
n n n
k k m m m m m m m m m k
in above equation, we have
1 2 1
1
1 1 1 1 2 2 2 2 ...
,..., 0 1 1 1 2 2 2
1 1 2 0 1 ; ; ; ; ; ; ... ; ; ; , , ... , . .. . ... ! ! !
n n nn n n n
m m m
m m n n n
m m k
n
n k k
n
B b m c b p B b m c b p B b m c b p
B b c b B b c b B b c b
x x t
m m m
m m k
1 2 1 1 1 21 1 1 1 2 2 2 2
...
,..., 0 1 1 1 2 2 2
... 1 1 ; ; ; ; ; ; ... ; ; ; , , ... , ( ) ( )
. ... 1
! !
n n n nn n n n m m m
m m n n n
m m
m m m n
n
B b m c b p B b m c b p B b m c b p
B b c b B b c b B b c b
x x t m m
1
,
1,
2,...,
;
1,
2,...,
;
1,
2,...,
; ;
nA n n n
t
F
b b
b c c
c x x
x
p
Theorem-2:For
R p
0,R
0,C&t 1
1 2 1 2 1 2
0
, , ,..., ; , ,..., ; , ,..., ; ; n
A n n n
k k
F k b b b c c c x x x p
1
2
1 2 1 2
1
, ,
,...,
; ,
,...,
;
,
,...,
; ;
1
1
1
n n
A n n
x t
x t
x t
t
F
b b
b c c
c
p
t
t
t
Proof: using above definition of Lauricella function (8) and changing order of summation, we get
1
1 1 1 1 2 2 2 2
,..., 0 1 1 1 2 2 2
;
; ;
;
; ;
...
;
; ;
.
,
,
....
,
n
n n n n
m m n n n
B b
m c
b p
B b
m c
b p
B b
m c
b
p
B b c
b B b c
b
B b c
b
1
1 1 ... 0 1 . ... ! ! ! n n
m m k
n
k m m
k n
x x t
k
m m k
Using following factorial function property in above theorem
1 2 1 2 ... ... 1 2 1 ! [ ... ]! n nm m m
m m m
n k
k k m m m
, we get
1 1 1 21 1 1 1 2 2 2 2
,..., 0 1 1 1 2 2 2
... 1
0 1 1 2
; ; ; ; ; ; ... ; ; ; , , ... , 1 ... ! ! [ ... ]! n n n
n n n n
m m n n n
m m k
m m m n
k
k n n
B b m c b p B b m c b p B b m c b p
B b c b B b c b B b c b
x x t
m m k m m m
If we put, k k m1m2....mn,we get the required result.
Corollary:
1 2 1 2 1 2
0
1 1 1 2 2 2
0, 0, & 1
,
, ,
,...,
; ,
,...,
; ,
,...,
; ;
; ;
,
;
;
,
...
;
;
,
n k
A n n n
k
t
p p p n n n
R p R C t
for
F
k b b
b c c
c x x
x p
t
e
b c
x t
b c
x t
b c
x t
Proof: By replacing k
in the theorem (2), then we have
1 2 1
1 2
1 1 1 1 2 2 2 2
...
,..., 0 1 1 1 2 2 2
1 2 1 2 ; ; ; ; ; ; ... ; ; ; 1 , , ... , /
/ 1 / 1 1
....
(1 / ) ! (1 / ) ! (1 / )
n n
n
n n n n m m m
m m n n n
m m m
n
n
B b m c b p B b m c b p B b m c b p
t
B b c b B b c b B b c b
x t
x t x t
t m t m t m
!
1 1 2
1 2 1 1 2 ... ... ... ,..., 0
1 1 1 1 2 2 2 2
1 1 1 2 2 2
1 2
1
1
;
; ;
;
; ;
...
;
; ;
.
,
,
...
,
.
...
,
n n n n nm m m m m
m m m m m
n n n n n n n
m m m
n
t
B b
m c
b p
B b
m c
b p
B b
m c
b p
B b c
b B b c
b
B b c
b
x t
x t
x t
usingbinomial theorem,taking limit , and using the property
1 2 ... 1 2 1 2 ...
( )
(
...
)
(
)
( )
n n
m m m
m
m
m
n kk
m m m k
and
/
,
0
Lim x n xn n N
n
, we get desired result.
3. SPECIAL CASES
Case-I: if we put
1 in theorem 1 and theorem 2 above results are reduced to the extended Lauricella function.
( )
1 1 1
0
1 2
1 1
{ 0, 1}
, ,.... ; ,... ; ,... ;
!
1
, ,.... ; ,... ;
,
...,
;
,
1
1
1
.
k n
A n n n
k k
n n
A n n
R p C and t
t
F
k b
b c
c x
x p
k
x
x
x
t
F
b
b c
c
p
t
t
t
where
and
1 2 1 2 1 2
0
1 2
1 2 1 2
0,
&
1 ,
, ,
,...,
; ,
,...,
; ,
,...,
;
!
1
, ,
,...,
; ,
,...,
;
,
,...,
;
.
1
1
1
k n
A n n n
k k
n n
A n n
for R p
C
t
t
F
k b b
b c c
c x x
x
p
k
x t
x t
x t
t
F
b b
b c c
c
p
t
t
t
Case-2: If we put1,p0, in theorem 1 and 2, the above results arereducedtotheLauricella function.
Case-3: If we put
x
3
x
4
...
x
n
0
,then results are converted into generalised extended second Appell function.Case-4: If we put
x
2
x
3x
4
...
x
n
0
then results are true for generalised extended Gauss hypergeometric function.4.CONCLUSION
REFERENCES
[1]M. A. Chaudhry, S. M. Zubair, Generalized incomplete gamma functions withapplications,J. Comput. Appl. Math. 55 (1994), 99–124. [2] M. A. Chaudhry, A. Qadir, M. Rafique, S. M. Zubair, Extension of Euler’s betafunction, J. Compt. Appl. Math. 78 (1997), 19–32. [3] M. A. Chaudhry, A. Qadir, H. M. Srivastava, R. B. Paris, Extended hypergeometricand confluent hypergeometric function, Appl. Math.Comput. 159 (2004),589–602.
[4] M.A. Chaudhry, S.M. Zubair, On a Class of Incomplete Gamma Functions withApplications, CRC Press (Chapman and Hall), Boca Raton, FL, 2002.
[5]M. A. Chaudhry, S. M. Zubair, Extended incomplete gamma functions with applications,J. Math. Anal. Appl. 274 (2002), 725–745. [6] E. Ӧzergin, M.A. Ӧzarslan , A. Altin, Extension of gamma, beta and hypergeometricfunction, J. Comp. and Appl. Math, 235(16)(2011), 4601–4610.
[7] M.A. Ӧzarslan, E. Ӧzergin, Some generating relations for extended hypergeometricfunction via generalized fractional derivative operator. Math.Comput.Model.52(2010), 1825–1833.
[8]E.D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted ByChelsea Publishing Company, Bronx, New York, 1971.
[9]D. M. Lee , A. K. Rathie, R. K. Parmar, Y. S. Kim, Generalization of ExtendedBeta Function, Hypergeometric and Confluent Hypergeometric Functions, HonamMath. Journal 33 (2) (2011), 187–206.
