On Two Extension Formulas for Lauricella’s Function of the Second Kind of Several Variables

http://dx.doi.org/10.4236/jamp.2016.43062

On Two Extension Formulas for Lauricella’s

Function of the Second Kind of Several

Variables

Ahmed Ali Atash

1

, Ahmed Ali Al-Gonah

2

1_{Department of Mathematics, Faculty of Education-Shabwah, Aden University, Aden, Yemen}

2_{Department of Mathematics, Faculty of Education-Aden, Aden University, Aden, Yemen}

Received 1 February 2016; accepted 27 March 2016; published 30 March 2016

Copyright © 2016 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract

The aim of this research paper is to derive two extension formulas for Lauricella’s function of the second kind of several variables with the help of generalized Dixon’s theorem on the sum of the series ₃F₂

( )

1 obtained by Lavoie et al. [1]. Some special cases of these formulas are also deduced.

Keywords

Extension Formulas, Lauricella’s Function, Dixon’s Theorem, Hypergeometric Functions

1. Introduction

The Lauricella’s function F_B( )r is defined and represented as follows [2]

( )

₍

₎

( )

( ) ( )

( )

( )

1

1 1

1 ₁

1 1 1

1 1

1

, , 0 1

, , , , , ; ; , ,

! !

r

r r

r _r

r

B r r r

m m

r r

m m m m _r

m m m m r

F a a b b c x x

a a b b _{x x}

c m m

∞

= _{+ +}

=

∑

 _

  

 



(1.1)

{

1

}

max x ,,xr <1;

where

( )

n

( )

1,

₍

₎₍

_{) (}

₎

if 0

1 2 1 , if 1, 2,3,

n

n a

a a a a n n

= 

=  _{+ + + − =}

   (1.2)

(

)

( )

, 0, 1, 2, .

a n a a

Γ +

= ≠ − −

Γ  (1.3)

Also, we note that

(

)

1 1 1 1

1 2 1

2 2 2 2

a

a a a

     

Γ_{ }Γ + = Γ_ + _{ }Γ + _

      (1.4)

( )

2

2

1 1 1

2

2 2 2

n n

n n

a = _ a _{ } a+ _

    (1.5)

(

)

( )

(

1

( )

1

)

n

n

a n

a a

Γ − −

=

Γ − (1.6)

( )

2 1

2 ! 2 !.

2

n n

n =  _{ } n

  (1.7)

The generalized Lauricella’s function of several variables is defined as follows [2]

( )

( )

[

]

( )

( )

( )

( )

_{( )}

( )

( ) ( )

( )

( )

_{( )}

( )

( ) ( )

(

)

1

1

1

1

1 1

, , 0 1

: ; ;

, ,

: ; ;

: , , : : ; ; : ;

: ; ;

, ,

: ; ; : , , : : ; ; : ;

, , ,

! !

n

n

n

n n

n n n

n

n

n _{n n n}

m m

n n

m m n

A B B

F z z

C D D

a b b

A B B

F z z

C D D c d d

z z

m m

m m

θ θ φ φ

ψ ψ δ δ

∞

= ′ ′



  ′  _ ′ ′_  

′      

≡ _{ }

 

′ __ ′ _ _ ′ ′_{  } 



=

∑

Ω

 

 

 





  

 

(1.8)

where

(

)

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

1 1

1 1

1 1 1

1

'

1 1 1

, ,

n

n n

j n j j n j

n

n _n

j n j j n j

A B B

n

j _{m m} j _m j

m

j j j

n

C D D

n

j _{m m} j _m j

m

j j j

a b b

m m

c d d

θ θ φ φ

ψ ψ δ δ

′

′+ + ′

= = =

′+ + ′

= = =

′

Ω =

′

∏

∏

∏

∏

∏

∏





 



(1.9)

the coefficients θ( )jk ,j=1, 2, ;Aφ( )jk ,j=1, 2,,B( )k ;ψ( )jk ,j=1, 2,, ;C δj( )k ,j=1, 2,,D( )k , for all

{

1, 2, ,

}

k∈  n are real and positive;

( )

a abbreviates the array of A parameters; a₁,aA,

( )

b( )k abbreviate the array of B( )k parameters b( )jk ,j=1, 2,,B( )k for all k∈

{

1, 2,,n

}

with similar inter pretations for

( )

c and

( )

d( )k k∈1, 2,, ;n et cetera. Note that, when the coefficients in Equation (1.8) equal to 1, the genera-lized Lauricella function (1.8) reduces to the following multivariable extension of the Kamp’e de F’eriet func-tion [2]:

[

]

( )

( )

( )

( )

( )

( )

( )

( )

(

)

1

1

1

1

1 1

1 1

1 1

1 1

, , 0 1

: ; ; : ; ; : ; ; ;

, , , ,

: ; ; : ; ; : ; ; ;

, , ,

! !

n

n

n

n

n q q

p

n n

n n

n

n n _{l m}

m

s s

n n

s s n

b b

a

p q q p q q

F z z F z z

l m m l m m _{c d d}

z z

s s

s s

∞

=

 _′ 

 

≡ _{ }

′

 

 

=

∑

Ω



  

 

  

 

where

(

)

( )

( )

( )

( )

( )

( )

( )

( )

1

1 1

1

1 1

1 1 1

1

1 1 1

, ,

n

n n

n

n n

q q

p

n j _{s s} j _s j

s

j j j

n l m m

n j _{s s} j _s j

s

j j j

a b b

s s

c d d

+ +

= = =

+ +

= = =

′

Ω =

′

∏

∏

∏

∏

∏

∏





 



. (1.11)

In the theory of hypergeometric series, classical summation theorems such as Dixon, Watson and Whipple for the series ₃F₂, have many generalizations and wide applications; see for example [1] [3]-[6]. In the present in-vestigation, we shall require the following generalization of the classical Dixon’s theorem for the series ₃F₂

( )

1 [1]:

(

) (

)

(

)

( ) ( ) (

) (

)

3 2

2

,

, , ;

1

1 ,1 ;

1 1 1

2 1 1

2 2 2

1 2 1

1 1 1 1 1

1

2 2 2 2 2

1 1 1

2 2 2

c i j

i j

a b c F

a b i a c i j

a b i a c i j b i i c i j i j b c a c i j a b c i j

i j j

a c a b c i A

a − + +

 

 _{+ − + + − + +} 

 

   

Γ + − + Γ + − + + Γ_ − − _{ }Γ − + + + _

   

=

Γ Γ Γ + − + + Γ + − − + +

  + +    +  Γ_ − + +_{  }Γ − − + + +_{ }

   

   

×

 

Γ_ + _Γ

 

,

1 2

1 1 3

1

2 2 2 2 2

,

1 1 1 1

2 2 2 2

i j

i a b

i j j

a c a b c i B

i a a b

  

  

 _{ − + + }

 

 _{  }





  +    

Γ_ − + +_{  }Γ − − + + + _   

   

   

+ _

 + 

    _

Γ_{  }Γ − + +_{  }

     

(1.12)

(

)

(

)

2 2 2 2 ;

3, 2, 1, 0,1, 2; 0,1, 2, 3

R a b c i j

i j

− − > − − −

= − − − = ,

where

[ ]

x denotes the greatest integer less than or equal to x and x denotes the usual absolute value of x. The coefficients A_{i j,} andB_{i j,} are given respectively in [1]. When i= =j 0, (1.12) reduces immediately to the classical Dixon's theorem [3], (see also [6])

(

) (

)

(

)

(

)

3 2

, , ;

1

1 , 1 ;

1 1

1 1 1 1

2 2

1 1

1 1 1 1

2 2

a b c F

a b a c

a a b a c a b c

a a b a c a b c

 

 _{+ − + −} 

 

   

Γ +_{ }Γ + − Γ + − Γ +_ − − _

   

=

   

Γ + Γ +_ − _{ }Γ + − Γ + − −_

   

(1.13)

(

)

{

R a−2b−2c > −2 .

}

2. Extension Formulas

( )

₍

₎

(

)

(

) (

)

(

)

( )

(

) (

)

(

)

{

( )

₍

₎

( )

₍

₎

}

1 1 1 1 ₁ 2

1 1 1 1 1 1

2 2

1 2 2 1 2 2 1

0 0 2 2 1

1 1

1 1 1 1 , 1 1 1 1 , 1 1 1 1

, , , , , , , , , ; ; , , , ,

2 ! 2 !

, , 2 , , , , 2 , , , , 2 , ,

r

r r

r _r

r

B r r r r r r

m m

r r r

m m m m

m m m m r

i j i j

F a i a a i a b i j b b i j b c x x x x

a i a i b i j b i j x x

c m m

I a b m i j A A a b m i j B B a b m i j

I ∞ ∞ = = _{+ +} − − − − − − − − − − − − − − = × + × ×

∑ ∑

         

(

)

{

( )

₍

₎

( )

₍

₎

}

(

)

(

)

(

)

(

)

( )

(

) (

)

(

)

( )

₍

₎

1 1 1 1 ₁ , ,

2 1 2 1

1 2 1 2 1 1 2 1 2 1 1

0 0 2 1 2 1 1

1

1 1 1 1 , 1 1 1 1 ,

, , 2 , , , , 2 , , , , 2 , ,

2 1 ! 2 1 !

, , 2 1, , , , 2 1, ,

r

r r

r _r

r r

r r r r i j r r r r i j r r r r

m m

r r r

m m m m

m m m m r

i j i j

a b m i j A A a b m i j B B a b m i j

a i a i b i j b i j x x

c m m

I a b m i j C A a b m i j D

+ + ∞ ∞ _{+ + + +} = = _{+ + + +} + + − − − − − − + + + × + + +

∑ ∑

       ( )

₍

₎

{

}

(

)

{

( )

₍

₎

( )

₍

₎

}

1

1 1 1 1

, ,

, , 2 1, ,

, , 2 1, , r , , 2 1, , r , , 2 1, ,

r r r r i j r r r r i j r r r r

B a b m i j

I a b m i j C A a b m i j D B a b m i j

+ × × + + + +  (2.1) and ( )

₍

₎

( ) (

)

(

) ( ) (

)

(

)

( )

(

) (

)

(

)

( )

₍

₎

1 1 1 1 ₁ 2 1

1 1 1 1 1 1

2 2

1 2 2 1 2 2 1

0 0 0 2 2 1

1

1 1 1 1 , 1 1 1 1

, , , , , , , , , , , ; ; , , , , ,

! 2 ! 2 !

, , 2 , , , , 2 , ,

r

r r

r _r

r

B r r r r r r

m m

m

r r r

m m m m m m

m m m m m m r

i j

F a a i a a i a b b i j b b i j b c x x x x x

a a i a i b b i j b i j x x x

c m m m

I a b m i j A A a b m i j B + ∞ ∞ ∞ = = = _{+ + +} − − − − − − − − − − − − − − = × +

∑ ∑ ∑

         ( )

₍

₎

{

}

(

)

{

( )

₍

₎

( )

₍

₎

}

( ) (

)

(

)

( ) (

)

(

)

(

) (

)

1 1 1 1 1 1

, 1 1 1 1

, ,

1 2 1 2 1

0 0 0 2 1 2 1

2 1 2 1

1 2 1 2 1 1

1

1

, , 2 , ,

, , 2 , , , , 2 , , , , 2 , ,

( )

! 2 1 ! 2 1 !

r r r r r i j r r

r r r r i j r r r r i j r r r r

r

m m m

m m m m m m

m m

m

r r

m m m

r

B a b m i j

I a b m i j A A a b m i j B B a b m i j

a a i a i

c

b b i j b i j x x x

m m m

I a ∞ ∞ ∞ + + = = = + + + + + + + + + × × + + − − + − − − − × + + ×

∑ ∑ ∑

       

(

)

{

( )

₍

₎

( )

₍

₎

}

(

)

{

( )

₍

₎

( )

₍

₎

}

1 1

1 1 1 , 1 1 1 1 , 1 1 1 1

, ,

, , 2 1, , , , 2 1, , , , 2 1, ,

, , 2 1, , , , 2 1, , , , 2 1, ,

i j i j

r r

r r r r i j r r r r i j r r r r

b m i j C A a b m i j D B a b m i j

I a b m i j C A a b m i j D B a b m i j

+ + + + × × + + + +  (2.2) where

(

)

(

) (

)

(

)

( ) ( ) (

) (

)

2

, , , , 2 1 1

1 1 1

2 2 2

1 2 1

r b i j

r r r r r r r r

r r

r r r r r r r I a b m i j m a i m b i j

a i i b i j i j

a b m b i j m a b i j − + +

= Γ − − + Γ − − + +

   

Γ_ − − _{ }Γ − + + + _

   

×

Γ Γ Γ − − + + Γ − − − + +

(2.3)

(

)

1 1 1 1 1

1

2 2 2 2 2

, , , ,

1 1 1

1

2 2 2 2

r r r r r

r r r r

r r r

i j j

m b m a b i

A a b m i j

i m m a

  + +    + 

Γ_ − − +_{  }Γ − − − + +_{ }

   

   

=

 

   

Γ_ − _{ }Γ − − +_{ }

    

(

)

1 3 1

1

2 2 2 2 2

, , , ,

1 1 1 1

2 2 2 2

r r r r r

r r r r

r r r

i j j

m b m a b i B a b m i j

i m m a

  +    

Γ −_ − +_{  }Γ − − − + +_{ }

   

   

=

 + 

   

Γ −_{  }Γ − − +_{ }

    

(2.5)

for i= − − −3, 2, 1, 0,1, 2; j=0,1, 2, 3;r=1, 2, 3,.

The coefficients A_{i j}( )_,r ,B_{i j}( )_,r ,C_{i j}( )_,r and D_{i j}( )_,r can be obtained from the tables of A_{i j,} and B_{i j,} given in [1] by replacing a by −2mr and −2mr −1 respectively.

Proof of (2.1): Denoting the left hand side of (2.1) by S, expanding F_B( )2r in a power series and using the results [2]:

( )

a m n+ =

( ) (

a m a+m

)

n (2.6)

(

)

(

)

0 0 0 0

, ,

m m n m n

A n m A n m n

∞ ∞ ∞

= = = =

= −

∑ ∑

∑ ∑

(2.7)

( )

₍

( ) ( )

1

₎

,0 1 n m m n n a

a n m

a m −

−

= ≤ ≤

− − and

(

) ( )

( )

1 !

! , 0

n

n

m

m n n m

m −

− = ≤ ≤

− , (2.8)

we get

(

)

(

) (

)

(

)

( )

(

)

(

)

1 1 1 1 ₁

1 1 1

0 0 1

1 1 1 1

! ! , , , , , , , , r r r r _r m m

r r r

m m m m

m m m m r

r r r r

a i a i b i j b i j x x S

c m m f a b m i j f a b m i j

∞ ∞ = = _{+ +} − − − − − − = × × ×

∑ ∑

       (2.9) where

(

)

3 2

, , ;

, , , , 1 .

1 ,1 ;

r r r r r r r

r r r r m a b f a b m i j F

m a i m b i j

−

 

= _{ }

− − + − − + +

  (2.10)

Separating (2.9) into its even and odd terms, we have

(

)

(

) (

)

(

)

( )

(

)

(

)

(

)

_{( )}

(

)

(

)

(

)

(

)

(

)

1 1 1 1 ₁ 1 2

1 _{1 2}

1 2

2 2

1 2 2 1 2 2 1

0 0 2 2 1

1 2 1 2 2 2 1 1 1 1

0 0 _{2 1 2 2}

1 2 1 2 2 2 1

2 ! 2 !

, , 2 , , , , 2 , ,

r r r r _r r r _r r m m

r r r

m m m m

m m m m r

r

m m m

r r r r

m m _{m m m}

r

m m m

a i a i b i j b i j x x S

c m m

a i a i a i f a b m i j f a b m i j

c b i j b i j b i j x

∞ ∞ = = _{+ +} ∞ ∞ + = = _{+ + + +} + − − − − − − = − − − × × × + − − − − − − ×

∑ ∑

∑ ∑

          

(

) (

) (

)

(

)

(

)

(

)

(

)

_{( )}

(

)

(

)

(

)

(

)

(

)

1 2 1 1

1 _{1 1}

1 1

1 1

2 1 2 2

1 1 1 1

1 2

1 2 1 2 2 1

2 2 2 2

0 0 _{2 2 2 1}

2 2 2

1 2 1 2 2 1 1 1

, , 2 1, ,

2 1 ! 2 ! 2 !

, , 2 , , , , 2 , ,

r

r r

r _{r r}

r r

r r

m m m r r

r r

m m m

r r r r

m m _{m m m}

m m m

r r r r

m m m

x x

f a b m i j

m m m

a i a i a i f a b m i j f a b m i j

a b i j b i j b i j x x x

− − − − + ∞ ∞ ₋ + = = _{+ + + +} + − + − × + + − − − × × × + + − − − − − − ×

∑ ∑

        

(

) (

) (

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

( )

(

) (

)

(

)

1 1 1 1 ₁ 1 1 1

1 1 1 1 1 1 1 1

2 1 2 1 1 2 1 2 1 1 2 1 2 1 1

0 0 2 1 2 1 1

1 1 1 1

2 ! 2 ! 2 1 !

, , 2 , , , , 2 , , , , 2 1, ,

2 1 ! 2 1 !

, , 2 1, , , ,

r

r r

r _r

r r

r r r r r r r r

m m

r r r

m m m m

m m m m r

r r r m m m

f a b m i j f a b m i j f a b m i j a i a i b i b i x x

c m m

f a b m i j f a b − − − − − + + ∞ ∞ + + + + = = _{+ + + +} + × × × × + − − − − + + + × + × ×

∑ ∑

      



(

2mr+1, ,i j

)

Finally, in (2.11) if we use the result (1.12), then we obtain the right hand side of (2.1). This completes the proof of (2.1). The result (2.2) can be proved by the similar manner.

3. Special Cases

1) In (2.1), if we take i= =j 0 and use the results (1.3)-(1.7), then after some simplification we obtain the following transformation formula:

( )

₍

₎

(

) (

)

(

) (

)

2

1 1 1 1 1 1 2 2

1 1 1 1 1 1 1 1

2 2 1

, , , , , , , , , ; ; , , , , , ,

1 1

0 :4; ; 4 : , , , 1 ;

2 2

1 1 1

2 :1; ;1 , : ;

2 2 2

1 1

; , , 1 ;

, ,

2 2

; ;

r

B r r r r r r

r r r r r r

r r r

F a a a a b b b b c x x x x x x

a b a b a b F

c c

a b

a b a b a b

x x a b

      

− − −

+ + +

=

+ ₊



+ + + _



+ _

  

 



 

(3.1)

which for

r

=

1

, reduces immediately to a known result of Bailey [7]

[

]

(

) (

)

2

3 4 3

1 1

, , , 1

;

2 2

, , , ; ; ,

1 1 1 ;

, ,

2 2 2

a b a b a b

F a a b b c x x F x

c c a b

 _{+ + +} 

 

− =  

 _{+ +} 

 

 

(3.2)

where F₃ is Appell’s function [2].

2) Similarly, in (2.2) if we take i= =j 0 and use the results (1.3)-(1.7), then we obtain the following trans-formation formula:

( )

₍

₎

(

)

( ) ( ) (

) (

)

(

₍

)

₎

(

)

(

) (

)

(

)

2 1

1 1 1 1 1 1

1 1 1 1 1 1

1 1

, , , , , , , , , , , ; ; , , , , ,

1 1

:1 , :1 , :1 , 1 :1

0 :2; 4; ; 4 : :1 , :1 ; ;

2 2

:1, 2, , 2

1: 0;1; ;1 : ; ;

:1

1 1

:1 , :1 , :1 ,

;

2 2

; r

B r r r r r r

r r r r r

F a a a a a b b b b b c x x x x x

a b a b a b a b

F

c

a b

a b a b a

+

      

− −

 ₊   _{+ +} 

− − − − _{   }

=    

− − − − ₊

 ₊  ₊

 

 

  



 

 

(

)

(

)

2 2 1

1 :1 ;

, 4 , , 4 ;

:1

r

r r r

b

x x x a b



 ₊ 

  

  _



+ _



(3.3)

3) In (2.2) if we take

r

=

1

, then we get a known extension formulas [8] for Lauricella’s function of three va- riables FB( )3

(

a a, ₁−i a b b, , ,_{1 1}− −i j b c x x, ₁; ; , ₁,−x₁

)

for

{

i= − − −3, 2, 1,0,1, 2; j=0,1, 2,3

}

.

References

[1] Lavoie, J.L., Grondin, F., Rathie, A.K. and Arora, K. (1994) Generalizations of Dixon’s Theorem on the Sum of a ₃F₂.

Mathematics of Computation, 62, 267-276.

[2] Srivastava, H.M. and Manocha, H.L. (1984) A Treatise on Generating Functions. Hasted Press, New York. [3] Bailey, W.N. (1935) Generalized Hypergeometric Series. Cambridge University Press, Cambridge.

[4] Lavoie, J.L., Grondin, F. and Rathie, A.K. (1992) Generalizations of Watson’s Theorem on the Sum of a ₃F₂. Indian Journal of Mathematics, 34, 23-32.

[7] Bailey, W.N. (1953) On the Sum of Terminating ₃F₂

( )

1 . Quarterly Journal of Mathematics Oxford, 2, 237-240.

http://dx.doi.org/10.1093/qmath/4.1.237

[8] Atash, A.A. (2015) An Extension Formulas of Lauricella’s Functions by Applications of Dixon’s Summation Theorem.