The aim of this paper is to derive certain relations involving Srivastava’s hypergeometric functions H B in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities, which express the aforementioned H B - hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell hypergeometric functions. Other closely-related results are also considered briefly. Also, we have derived certain new integral representations for the H B - hypergeometric functions of three variables defined earlier by Lauricella [ 8 ] and Srivastava [14 ] .

1. International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org ||Volume 5 Issue 4|| April 2016 || PP.01-08
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Certain D - Operator for Srivastava B
H - Hypergeometric
Functions of Three Variables
Mosaed M. Makky
Mathematics Department, Faculty Of Science (Qena) South Valley University (Qena , Egypt)
Abstract : The aim of this paper is to derive certain relations involving Srivastava’s hypergeometric functions
B
H in three variables. Many operator identities involving these pairs of symbolic operators are first
constructed for this purpose. By means of these operator identities, which express the aforementioned B
H -
hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell
hypergeometric functions. Other closely-related results are also considered briefly. Also, we have derived
certain new integral representations for the B
H - hypergeometric functions of three variables defined earlier
by Lauricella [ 8 ] and Srivastava [14 ] .
Keywords: Decomposition formulas; Srivastava’s hypergeometric functions; Multiple hypergeometric
functions; Gauss hypergeometric function; Appell’s hypergeometric functions; Generalized hypergeometric
function.
2010 Mathematics Subject Classification: Primary 33C20, 33C65; Secondary 33C05, 33C60, 33C70.
I. Introduction
A great interest in the theory of multiple hypergeometric functions (that is, hypergeometric functions of several
variables) is motivated essentially by the fact that the solutions of many applied problems involving (for
example) partial differential equations are obtainable with the help of such hypergeometric functions (see, for
details, [17]; see also the recent works [11,12] and the references cited therein). For instance, the energy
absorbed by some nonferromagnetic conductor sphere included in an internal magnetic field can be calculated
with the help of such functions [12].
Hypergeometric functions of several variables are used in physical and quantum chemical applications as well
(cf. [14,16]). Especially, many problems in gas dynamics lead to solutions of degenerate second-order partial
differential equations which are then solvable in terms of multiple hypergeometric functions. Among examples,
we can cite the problem of adiabatic flat-parallel gas flow without whirlwind, the flow problem of supersonic
current from vessel with flat walls, and a number of other problems connected with gas flow [5].
We note that Riemann’s functions and the fundamental solutions of the degenerate second order partial
differential equations are expressible by means of hypergeometric functions of several variables [6]. In
investigation of the boundary-value problems for these partial differential equations.
The familiar operator method of Burchnall and Chaundy (cf. [2,3]; see also [4]) has been used by them rather
extensively for finding recurrence formulas for hypergeometric functions of two variables in terms of the
classical Gauss hypergeometric function of one variable.
Lauricella [8] actually defined the ten triple hypergeometric functions , , .....E F R
F F F in addition, of course,
to his four functions , ,A B C
F F F and D
F of three ( or n) variables, Srivastava [ 14 ] added three new
functions ,A B
F F and C
F to the Lauricella set of function hypergeometric functions of three variables, here we
shall obtain the integral representations of B
H - hypergeometric functions in quite a different form.
Suppose that a hypergeometric function in the form (c.f. [4,13])
(1.1)   n
n n
nn
z
n
zF 



1
12
)(!
)()(
1;;,



for  neither zero nor a negative integer.
Now we consider B
H - hypergeometric function defined in ([17] as follows :
(1.2) B
H =  1 2 3 1 2 3 1 2 3
, , ; , , ; , ,B
H z z z     

2. Certain D - Operator For Srivastava B
H - Hypergeometric Functions Of Three Variables
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= 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 3
1 2 3
, , 0 1 2 3 1 2 3
( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
z z z
n n n
  
  

  


which were introduced and investigated, over four decades ago, by Srivastava (see, for details, [14,15]; see also
[17, p. 43] and [18, pp. 68–69]). Here, and in what follows
 
 
 
 


 


denotes the Pochhammer symbol (or the shifted factorial) for all admissible (real or complex) values of λ and μ.
Also, we study the B
H - hypergeometric function, where it is regular in the unit hypersphere (c.f. [2,3]), for the
B
H - function, we can define as contiguous to it each of the following functions, which are samples by
uppering or lowering one of the parameters by unity.
(1.3) B
H ( +) = 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 31 1 3
1 2 3
, , 0 1 1 2 3 1 2 3
( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n n
z z z
n n n
  
   

  

 

(1.4) B
H ( -) = 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 31
1 2 3
, , 0 1 1 3 1 2 3 1 2 3
( ) ( ) ( )
( ) ( ) ( )
1 ! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
z z z
n n n n n
  
   

  
   

(1.5) B
H ( +,  +)
= 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 31 1 3 2 1 2
1 2 3
, , 0 1 2 1 2 3 1 2 3
( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n n n n
z z z
n n n
   
    

  

   
 ,
(1.6) 


3
1j
j
dD ,
j
jj
z
zd



and the way we effect it with the recursions relations as it is found in the second part of the research..
We obtain, as a result of acting by D on this function a differential equation, some special cases for a group of
differential equations are the functions that are effected by the differential operator. There is a numerical
example for one of these cases.
II. A Set Of Operator Of B
H - Hypergeometric Function
By applying the operator D in (1.6) to (1.2), we find the following set of operator identities involving the Gauss
function 2 1
F , the Appell functions, and Srivastava’s hypergeometric functions B
H defined by (1.2) is
(2.1) D  1 2 3 1 2 3 1 2 3
, , ; , , ; , ,B
H z z z     
= D 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 3
1 2 3
, , 0 1 2 3 1 2 3
( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
z z z
n n n
  
  

  


= 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 3 1 2 3
1 2 3
, , 0 1 2 3 1 2 3
( )( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n n n
z z z
n n n
  
  

  

 

= 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 1 2 1 3 1
1 2 3
, , 0 1 2 3 1 1 2 3
( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
z z z
n n n
  
  

     
 

+ 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 1 3 1 1
1 2 3
, , 0 1 2 3 1 2 1 3
( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
z z z
n n n
  
  

     
 

+ 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 1 2 3 1 1
1 2 3
, , 0 1 2 3 1 2 3 1
( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
z z z
n n n
  
  

     
 

1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 31 1 3 2 1 2
1 1 2 3
, , 0 1 1 1 2 3 1 2 3
( ) ( ) ( )( )( )
( ) ( ) ( ) ( )
( ) ! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n n n n
z z z z
n n n n
   
   

  

   




3. Certain D - Operator For Srivastava B
H - Hypergeometric Functions Of Three Variables
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1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 32 1 2 3 2 3
2 1 2 3
, , 0 2 2 1 2 3 1 2 3
( ) ( ) ( )( )( )
( ) ( ) ( ) ( )
( ) ! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n n n n
z z z z
n n n n
   
   

  

   



1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 31 1 3 3 2 3
3 1 2 3
, , 0 3 3 1 2 3 1 2 3
( ) ( ) ( )( )( )
( ) ( ) ( ) ( )
( ) ! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n n n n
z z z z
n n n n
   
   

  

   



1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 31 1 1 3 2 1 21 2
1 1 2 3
, , 01 1 2 1 1 1 2 3 1 2 3
( ) ( ) ( )( )( )
( ) ( ) ( ) ( )
( ) ! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n n n n
z z z z
n n n n
     
      

  

   



1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 32 3 2 2 1 2 3 2 3
2 1 2 3
, , 02 2 3 2 2 1 2 3 1 2 3
( ) ( ) ( )( )( )
( ) ( ) ( ) ( )
( ) ! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n n n n
z z z z
n n n n
      
      

  

   



1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 31 3 3 1 1 3 3 2 3
3 1 2 3
, , 03 1 3 3 3 1 2 3 1 2 3
( ) ( ) ( )( )( )
( ) ( ) ( ) ( )
( ) ! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n n n n
z z z z
n n n n
      
      

  

   



i.e.
(2.2) 1 2 1
1 2 3 1 2 3 1 2 3
1
( 1, 1, ; 1, , ; , , )B B
z
D H H z z z
 
     

   
2 3 2
1 2 3 1 2 3 1 2 3
2
( , 1, 1; , 1, ; , , )B
z
H z z z
 
     

   
1 3 3
1 2 3 1 2 3 1 2 3
3
( 1, , 1; , , 1; , , )B
z
H z z z
 
     

    .
From which and using the contiguous functions relations (1.3), (1.4), (1.5), (1.6) we have
(2.3)
2 3 11 2 1
1 2 1 2 3 2
1 2
1 3 1
1 3 3
3
( , ; ) ( , ; )
( , ; )
B B B
B
zz
D H H H
z
H
  
     
 
 
  

       
   
i.e. the partial differential equation
2 3 1 1 3 11 2 1
1 2 1 2 3 2 1 3 3
1 2 3
( , ; ) ( , ; ) ( , ; ) 0B B B B
z zz
D H H H H
    
        
  
 
             
 
has a solution in the form : -
 1 2 3 1 2 3 1 2 3
, , ; , , ; , ,B
H z z z      = 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 3
1 2 3
, , 0 1 2 3 1 2 3
( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
z z z
n n n
  
  

  

 .
Also, we can write the following recursion relations B
H - hypergeometric function as follows:
(2.4)  1 1 2 3 1 2 3 1 2 3
, , ; , , ; , ,B
d H z z z     
= 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 1 2 3
1 2 3
, , 0 1 2 3 1 2 3
( )( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n
z z z
n n n
  
  

  


= 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 1 2 1 3 1
1 2 3
, , 0 1 2 3 1 1 2 3
( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
z z z
n n n
  
  

     
 

1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 31 1 3 2 1 2
1 1 2 3
, , 0 1 1 1 2 3 1 2 3
( ) ( ) ( )( )( )
( ) ( ) ( ) ( )
( ) ! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n n n n
z z z z
n n n n
   
   

  

   



1 2 1
1 2 3 1 2 3 1 2 3
1
( 1, 1, ; 1, , ; , , )B
z
H z z z
 
     

   

4. Certain D - Operator For Srivastava B
H - Hypergeometric Functions Of Three Variables
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(2.5)  2 1 2 3 1 2 3 1 2 3
, , ; , , ; , ,B
d H z z z     
= 1 3 1 2 2 3 31 2
1 2 3 1 2 3
2 1 2 3
1 2 3
, , 0 1 2 3 1 2 3
( )( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n
z z z
n n n
  
  

  


= 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 1 3 1 1
1 2 3
, , 0 1 2 3 1 2 1 3
( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
z z z
n n n
  
  

     
 

1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 32 1 2 3 2 3
2 1 2 3
, , 0 2 2 1 2 3 1 2 3
( ) ( ) ( )( )( )
( ) ( ) ( ) ( )
( ) ! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n n n n
z z z z
n n n n
   
   

  

   



2 3 2
1 2 3 1 2 3 1 2 3
2
( , 1, 1; , 1, ; , , )B
z
H z z z
 
     

   
(2.6)  3 1 2 3 1 2 3 1 2 3
, , ; , , ; , ,B
d H z z z     
= 1 3 1 2 2 3 31 2
1 2 3 1 2 3
3 1 2 3
1 2 3
, , 0 1 2 3 1 2 3
( )( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n
z z z
n n n
  
  

  


= 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 1 2 3 1 1
1 2 3
, , 0 1 2 3 1 2 3 1
( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
z z z
n n n
  
  

     
 

1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 31 1 3 3 2 3
3 1 2 3
, , 0 3 3 1 2 3 1 2 3
( ) ( ) ( )( )( )
( ) ( ) ( ) ( )
( ) ! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
n n n n
z z z z
n n n n
   
   

  

   



1 3 3
1 2 3 1 2 3 1 2 3
3
( 1, , 1; , , 1; , , )B
z
H z z z
 
     

    .
III. Some Recurrence Relations For B
H - Hypergeometric Function
3.1 Putting 1 2
  and 1 2
  in (2.4), (2.5), and (2.6) we have
(3.1)  1 1 1 3 1 1 3 1 2 3
, , ; , , ; , ,B
d H z z z     
2
1 1
1 1 3 1 1 3 1 2 3
1
( 1, 1, ; 1, , ; , , )B
z
H z z z

     

   
(3.2)  2 1 1 3 1 1 3 1 2 3
, , ; , , ; , ,B
d H z z z     
1 3 2
1 1 3 1 1 3 1 2 3
1
( , 1, 1; , 1, ; , , )B
z
H z z z
 
     

   
(3.3)  3 1 2 3 1 2 3 1 2 3
, , ; , , ; , ,B
d H z z z     
1 3 3
1 1 3 1 1 3 1 2 3
3
( 1, , 1; , , 1; , , )B
z
H z z z
 
     

    .
3.2 Putting 1 3
  and 1 3
  in (2.4), (2.5), and (2.6) we have
(3.4)  1 1 2 1 1 2 1 1 2 3
, , ; , , ; , ,B
d H z z z     
1 2 1
1 2 1 1 2 1 1 2 3
1
( 1, 1, ; 1, , ; , , )B
z
H z z z
 
     

   
(3.5)  2 1 2 1 1 2 1 1 2 3
, , ; , , ; , ,B
d H z z z     
2 1 2
1 2 1 1 2 1 1 2 3
2
( , 1, 1; , 1, ; , , )B
z
H z z z
 
     

   

5. Certain D - Operator For Srivastava B
H - Hypergeometric Functions Of Three Variables
www.ijesi.org 5 | Page
(3.6)  3 1 2 1 1 2 1 1 2 3
, , ; , , ; , ,B
d H z z z     
2
1 3
1 2 1 1 2 1 1 2 3
1
( 1, , 1; , , 1; , , )B
z
H z z z

     

    .
3.3 Some special cases for the B
H - hypergeometric function which given us some differential equations
its functions as follows:
1- The function
 3 3 1 2 3
; ; , ,B
H z z z  = 2 3 31 2
1 2 3 3
3
1 2 3
, , 0 1 2 3 3
( )
( ) ( ) ( )
! ! !( )
n n nn n
n n n n
z z z
n n n





 .
Thus
D  3 3 1 2 3
; ; , ,B
H z z z 
= D 2 3 31 2
1 2 3 3
3
1 2 3
, , 0 1 2 3 3
( )
( ) ( ) ( )
! ! !( )
n n nn n
n n n n
z z z
n n n






= 2 3 31 2
1 2 3 3
1 2 3 3
1 2 3
, , 0 1 2 3 3
( )( )
( ) ( ) ( )
! ! !( )
n n nn n
n n n n
n n n
z z z
n n n





 

= 2 3 31 2
1 2 3 3
3 1
1 2 3
, , 0 1 2 3 3
( )
( ) ( ) ( )
! ! !( )
n n nn n
n n n n
z z z
n n n



 


+ 2 3 31 2
1 2 3 3
3 1 1
1 2 3
, , 0 1 2 3 3
( )
( ) ( ) ( )
! ! !( )
n n nn n
n n n n
z z z
n n n



  


+ 2 3 31 2
1 2 3 3
3 1 1
1 2 3
, , 0 1 2 3 3 1
( )
( ) ( ) ( )
! ! !( )
n n nn n
n n n n
z z z
n n n



  
 

2 3 31 2
1 2 3 3
3
1 1 2 3
, , 0 1 2 3 3
( )
( ) ( ) ( ) ( )
! ! !( )
n n nn n
n n n n
z z z z
n n n





 
2 3 31 2
1 2 3 3
3
2 3 2 3 1 2 3
, , 0 1 2 3 3
( )
( ) ( ) ( ) ( ) ( )
! ! !( )
n n nn n
n n n n
z n n z z z
n n n






  
2 3 31 2
1 2 3 3
33 2 3
3 1 2 3
, , 0 3 3 1 2 3 3
( )( )
( ) ( ) ( ) ( )
( ) ! ! !( )
n n nn n
n n n n
n n
z z z z
n n n n

 



 


 .
The function
(3.7)  3 3 1 2 3
; ; , ,B
H z z z  = 2 3 31 2
1 2 3 3
3
1 2 3
, , 0 1 2 3 3
( )
( ) ( ) ( )
! ! !( )
n n nn n
n n n n
z z z
n n n





 .
has a solution of the equation
(3.8)   3 3
1 3 2 3 3 1 2 3 3 3 1 2 3
3
( 1; ; , , ) ( 1; 1; , , ) 0B B B
z
D z H z H z z z H z z z

    

 
       
 
2- The function:
 1 2 3
, , ; , ,B
H z z z   = 31 2
1 2 3
1 2 3
, , 0
( ) ( ) ( )
nn n
n n n
z z z


 .
Thus

6. Certain D - Operator For Srivastava B
H - Hypergeometric Functions Of Three Variables
www.ijesi.org 6 | Page
D  1 2 3
, , ; , ,B
H z z z   = 31 2
1 2 3
1 2 3 1 2 3
, , 0
( )( ) ( ) ( )
nn n
n n n
n n n z z z


 
= 31 2
1 2 3
1
1 2 3
, , 0
( ) ( ) ( )
nn n
n n n
z z z



 + 31 2
1 2 3
1
1 2 3
, , 0
) ( ) ( )
nn n
n n n
z z z



 + 31 2
1 2 3
1
1 2 3
, , 0
( ) ( ) ( )
nn n
n n n
z z z




D  1 2 3
, , ; , ,B
H z z z   31 2
1 2 3
1 2 3 1 2 3
, , 0
( ) ( ) ( ) ( )
nn n
n n n
z z z z z z


    .
The function
 1 2 3
, , ; , ,B
H z z z   = 31 2
1 2 3
1 2 3
, , 0
( ) ( ) ( )
nn n
n n n
z z z


 .
has a solution of the equation
   1 2 3 1 2 3
( ) , , ; , , 0B
D z z z H z z z      
Now will given a numerical example for one of these differential equations that we get, by giving a numerical
value for constant numbers in any previous equation as the one used in (3.8) and we get an equation
representing the surface sphere equation, its solution is solution for the equation (3.8) after substituting the same
numerical value. This clarifies the idea of the study.
Example :
Here we shall take the equation in (3.8) as follows:
  3 3
1 3 2 3 3 1 2 3 3 3 1 2 3
3
(2 .3) ( 1; ; , , ) ( 1; 1; , , ) 0B B B
z
D z H z H z z z H z z z

    

 
       
 
which has a general solution in equation (3.8) we can consider that
(3.10)
3 3
1 2 3
1
1n n n
  

  
Substituting from (3.10) in the equation (3.8) we get :-
 1 2 3
1;1; , ,B
H z z z = 2 3
1 2 3 3
1 2 3
, , 0
(1)
( )( )( )
1!1!1!(1)
n n
n n n n
z z z



 and
since
1321
 nnn , )!1)...(3)(2(1)1( 3232

nnnn
and 3 3
(1) 1( 2 )(3)...( 1) !n
n  .
 1 1 2 3 2 1 2 3 3 1 2 3
(1;1; , , ) (1;1; , , ) (1;1; , , ) 0B B B
D z H z z z z H z z z z H z z z     
 1 2 3 1 2 3
(1;1; , , ) 0B
D z z z H z z z     
Applying that and the partial differential equation (3.9) we see that: -
0]0)([ 321
 A
HzzzD
0))](([ 321321
 zzzzzzD
 1 2 3 1 2 3 1 2 3 1 2 3 1 2
( ) ( )( ) ( ) 3 ( 0D z z z z z z z z z z z z z z z       
i.e. 03)( 321
 zzz
which is a surface equation in hypersphere has a solution in the form:
 1 2 3
1;1; , ,B
H z z z = 1 2 3
z z z

7. Certain D - Operator For Srivastava B
H - Hypergeometric Functions Of Three Variables
www.ijesi.org 7 | Page
IV. Integral Representations For Certain B
H -Hypergeometric Function Of Three Variables
Due To Lauricella And Srivastava
Now we consider B
H - hypergeometric function as follows
 1 2 3 1 2 3 1 2 3
, , ; , , ; , ,B
H z z z     
= 1 3 1 2 2 3 31 2
1 2 3 1 2 3
1 2 3
1 2 3
, , 0 1 2 3 1 2 3
( ) ( ) ( )
( ) ( ) ( )
! ! !( ) ( ) ( )
n n n n n n nn n
n n n n n n
z z z
n n n
  
  

  


and
(4.1)  1 2 3
1, 1, 1; 1, 1, 1; , ,B
H z z z                
=
           
     
3
1 1 1 1 1 1 2
1 1 1
     
     
      
    
           
        
     2 2 2
2 2 2
. cos cos cos
i i i
e
  
              
  
  
       
  
  
   
1
. 1 1 co s 2 sin 2 1 co s 2 sin 2i i x

   
 
      
   
1
. 1 1 co s 2 sin 2 1 co s 2 sin 2i i y

   
 
      
   
1
. 1 1 co s 2 sin 2 1 co s 2 sin 2i i z d d d

      
 
      
where
     R e 1 , R e 1 R e 1and              .
Proof :
 1 2 3
1, 1, 1; 1, 1, 1; , ,B
H z z z                
= 1 2 3
, , 0
( 1, )( 1, )( 1, )
! ! !( 1, )( 1, )( 1, )
m n p
m n p
m n p n m p
z z z
m n p m n p
     
  


        
  

=
           
     
1 1 1 1 1 1
1 1 1
     
     
           
        
1 2 3
, , 0
( 1, )( 1, )( 1, )
.
! ! !
m n p
m n p
m n p
z z z
m n p
  

  

   
   
 
   
1 1 1 ( 1)
.
1 1 1 1
m p p p m n
m p n m
     
   
              
           
   
   ( 1) 1 1( 1)
.
1 1 1
m n pn p
n p
   
 
          
     
Now, it is known that (Whittaker and Watsin [ ])
 
   
 2
2
1 2
cos
1 1
in
e d
 
    


 
  

 

  

   

where  R e 1   
so that the right hand side of ( 4.1) becomes
           
     
3
1 1 1 1 1 1 2
1 1 1
     
     
      
    
           
        

8. Certain D - Operator For Srivastava B
H - Hypergeometric Functions Of Three Variables
www.ijesi.org 8 | Page
     2 2 2
2 2 2
. cos cos cos
i i i
e
  
              
  
  
       
  
  
   
1
1
. 1 1 co s 2 sin 2 1 co s 2 sin 2i i z

   
 
      
   
1
2
. 1 1 co s 2 sin 2 1 co s 2 sin 2i i z

   
 
      
   
1
3
. 1 1 co s 2 sin 2 1 co s 2 sin 2i i z d d d

      
 
      
Using the identities
 2
2 co s 2 co s sin co s
1 co s 2 sin 2
i
e i
i

   
 
 
  
and
2 co s 1 co s 2 sin 2
i
e i

  

  
Then we have
 1, 1, 1; 1, 1, 1; , ,B
H x y z                 =
           
     
3
1 1 1 1 1 1 2
1 1 1
     
     
      
    
           
        
     2 2 2
2 2 2
. cos cos cos
i i i
e
  
              
  
  
       
  
  
   
1
1
. 1 1 co s 2 sin 2 1 co s 2 sin 2i i z

   
 
      
   
1
2
. 1 1 co s 2 sin 2 1 co s 2 sin 2i i z

   
 
      
   
1
3
. 1 1 co s 2 sin 2 1 co s 2 sin 2i i z d d d

      
 
      
References
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Applications, J. Phys. A: Math. Gen. 18 (1985) L227–L234.
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