Study of a Generating Function Involving Generalised Lauricella Function

(1)

STUDYOF A GENERATING FUNCTION

INVOLVING GENERALISED

LAURICELLA FUNCTION

Ekta Mittal

1

, Sunil Joshi

2

,Rupakshi Mishra Pandey

3

1_{Department of Mathematics, The IIS University, Jaipur, Rajasthan,India} 2_{Department of Mathematics & Statistics, Manipal University Jaipur, Rajasthan ,India}

3

Department of Applied Mathematics, Amity University Uttar Pradesh, Noida, India

[email protected], [email protected], [email protected]

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 cases

Key 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

e

dt

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

;

x

exp

,

(Re

> 0, Re

> 0) .

x p

t

pt

dt

x

p



 

(2)

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

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 2 2 2 2

...

,..., 0 1 1 1 2 2 2

1 1

;

; ;

;

; ;

...

;

; ;

;

...

;

...

,

!

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

m





 

































where



x₁  ... x_n 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

(3)

Theorem-1: The following function hold true

 

_

_





 



 



1 1 1

0

1 2

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

t

F

b

b c

c

p

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 n

A n n n

k k

t

F

k b

b c

c x

x p

k





 





 

0

!

k k k

t

k



 

 







 





 



   

1 1 1

1 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



 















  



   

1

1 1 ... 0 1

.

...

!

n n

m m _k

n

k m m

k n

x

t

k

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 n

n 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 2

1 1 1 1 2 2 2 2

...

,..., 0 1 1 1 2 2 2

... 1 1 ; ; ; ; ; ; ... ; ; ; , , ... , ( ) ( )

. ... 1

! !                   _ _ _    _ 



n n n n

n n n n m m m

m m n n n

m m

m m m n

n

x x t m m 







1 



,

1

,

2

,...,

;

1

,

2

,...,

;

1

,

2

,...,

; ;



n

A n n n

t



F



b b

b c c

c x x

x

p



(4)

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

n n

A n n

x t

t

F

b b

b c c

c

p

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 n

m m m

n k

k k m m m

  

 

 _



    , we get



 





 



   

 

_

_

1 1 1 2

1 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

x x t

m m k m m m





                      _{ } _ _{ }  



If we put, k k m₁m₂....m_n,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

t

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 n

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



(5)

usingbinomial theorem,taking limit   , and using the property

1 2 ... 1 2 1 2 ...

( )

(

...

)

(

)

( )

n n

m m m

m

n k

k

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 .

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

t

F

b

b c

c

p

t

where







  







 

_

_









and

 





 









_

_{ }

_

_

_

1 2 1 2 1 2

0

1 2

1 2 1 2

0,

&

1 ,

, ,

,...,

; ,

,...,

; ,

,...,

;

!

1 , ,

,...,

; ,

,...,

;

,

,...,

;

.

1

k n

A n n n

k k

n n

A n n

for R p

C

t

F

k b b

b c c

c x x

x

p

k

x t

t

F

b b

b c c

c

p

t























 



_

_



_







Case-2: If we put1,p0, 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

₃

x

₄

 

...

x

_n



0

then results are true for generalised extended Gauss hypergeometric function.

4.CONCLUSION

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REFERENCES

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