Recently,we introduced “An unification of certain generalized Geometric polynomial Set T n (x1 ,x2 ,x3 ,x4 ) ,with the help of generating function which contains Appell function of four variables in the notation of Burchnall and Choundy [4] associated with Lauricella function. This generated hypergeometric polynomial Set covers as many as thirty four orthogonal and non -orthogonal polynomials.In the present paper an attempt has been made to express a Triple finite integral representation of the polynomial set T n (x1 ,x2 ,x3 ,x4 ).

1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 6 Ver. 1 (Nov. - Dec. 2015), PP 09-14
www.iosrjournals.org
DOI: 10.9790/5728-11610914 www.iosrjournals.org 9 | Page
Finite Triple Integral Representation For The Polynomial Set
Tn(x1,x2,x3,x4)
Ahmad Quaisar Subhani, Brijendra Kumar Singh,
Ahmad Quaisar Subhani, Dept. of Maths,Maulana Azad College of Engg and technology,Anisabad,Patna,
Brijendra Kumar Singh,Department of Mathematics, J.P. University, Chapra,India
Abstract:-Recently,we introduced “An unification of certain generalized Geometric polynomial Set Tn
(x1,x2,x3,x4) ,with the help of generating function which contains Appell function of four variables in
the notation of Burchnall and Choundy [4] associated with Lauricella function. This generated
hypergeometric polynomial Set covers as many as thirty four orthogonal and non-orthogonal
polynomials.In the present paper an attempt has been made to express a Triple finite integral
representation of the polynomial set T
n
(x1,x2,x3,x4).
Key words:-Appell function, Lauricella form, Generalized hypergeometric polynomial.
I. Introduction
The generalized polynomial set Tn (x1,x2,x3,x4)is defined by means of generating relation
11
mt d
e
 
32 432 4
1 2 42 3 4
( ); ( ); ( )
, , ,
( );( );( )
r u i
mm mmm m
s v i
A C Eg
F x t x t x t x t
B D Fq

 
 
 
 
 
   
     
 1 2 3 4
1 2 3 4
: : : : : : :
1 2 3 4, : : : : : ( ) : ( ) : ( )
0
, , ,


  r u gi
s v i
A C E n
n m m m m m B D Fq
n
T x x x x t
    
… (1.1)
where (i = 1, 2, 3, 4) and 1 2 3 4, , , ,     are real and 1 2 3 4, , , ,m m m m m are positive
integer. The left hand side of (1.1) contains the product of two generalized hypergeometric function which
contains Appell function of four variables in the notation of Burchanall and Chaundy [4] associated with
Lauricella function. The polynomial set contains a number of parameters for simplicity, it is denoted by Tn
(x1,x2,x3,x4), where n is the order of the polynomial set.
After little ,simplification(1.1)gives,
1 2 3 4( , , , )nT x x x x
1 1 2 2 1 1 2 2 3 31 1
31 2 4
1 2 3 40 0 0 0
          
      
       
   
    
n m P m P n m P m P m Pn n m P
mm m m
P P P P
 
 
1 1 2 2 3 3 4 4
1 1 2 2 3 3 4 4
( 1) ( 1) ( 1)
( 1) ( 1) ( 1)
( )
( )
      
      

r n m P m P m P m P
s n m P m P m P m P
A
B
     
       
1 2 3
1 2 3
1 2 3
1 2 3
1 1 2 2 3 3 4 4
1 1 2 2 3 3 4 4
( ) ( ) ) )            

                
u g g gP P P
v q q q
P P P
C n m P m P m P m P E E E
D n m P m P m P m P F F F
         
 
3 41 1 2 2 3 3 4 4 1 32 41
4
4
2 2
4
4
1 1 3 42 3 4
1 2 3 42! ! ! !
   

       
 
 
 
P Pn m P m P m P m P P mP mPm
g
P
m P
q
P
E x x x
F P P x P P
…. ... (1.2)

2. Finite Triple Integral Representation For The Polynomial Set Tn(x1,x2,x3,x4)
DOI: 10.9790/5728-11610914 www.iosrjournals.org 10 | Page
The polynomial Set Tn (x1,x2,x3,x4) happens to the generalization of as many thirty four orthogonal and non-
orthogonal polynomials.
Notations
a)
I. (m)=1.2.3…….m
II. (A
p
)=A
1
.A
2
.A
3
.…….A
p
III. [(Ap)]=A1,A2,A3…….Ap
IV. [(Ap)]n=(A1)n,(A2)n,(A3)n…….(Ap)n
V. 1 1
( , ) , , ......
b b b a
a b
a a a
  
 
b)
I.
     
   
( ) ( )
( ) ( ) !
n
m
r un n
s vn n
A C x
M
B D n


2 Theorem : 2 31, 1m m  and 4 1m  , we have
 1 2 3 4, , ,nT x x x x
 
     
1 1 1 1 1 11
0 0 0
1 a b ca b c M
a b c
       
   
     
1 2 3 4
1 2 3 4
1 2 3 41 v : : : :
: : : :
[ : , , , ], [( 1) :1],
___________, [( ) :1], [( ) :1], [( ) :1]
s g g g g
r u q q q q
n m m m m a b c
F
a b c
 

   


   1 2 3 4 v 1 2 3 4(1 ( ) ) : , 1, 1, 1 , (1 ( ) ) : , , , ,sB n m m m m D n m m m m      
   1 2 3 4 1 2 3 4(1 ( ) ) : , 1, 1, 1 , (1 ( ) ) : , , ,r uA n m m m m C n m m m m       ,
   
 
1
1 2 3 4
1
1 2 3 4
( 1)
1
1
( ) :1 , ( ) :1 , ( ) :1 , ( ) :1 1
;
( ) :1 , ( ) :1 , ( ) :1 , ( ) :1
m r s u v
g g g g
m
m
q q q q
E E E E
F F F F x
   

               
             
,
 
 
2 1 2
2
( 1)
2
1 2
1
,
m r s u v r r
m
m
x x
     
 

 
 
33
3
( 1)
3 3
1
1
,
m r s u v r sm
m
m
x
x
     
 

 
 
44
4
( 1)
4 4
1
1
m r s u v r sm
m
m
x
x
      
  

 
d d d   …… (2.1)
Proof :-We have
I
1 1 2 21 1 1 1 2 2 3 3
31 2 4
1 2 3 4
1 1 1 1 1 1
0 0 0
0 0 0 0
n m P m Pn n m P n m P m P m P
mm m m
a b c
P P P P
          
      
           
   
         
   
   
1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4
1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4
( 1) ( 1) ( 1)
v( 1) ( 1) ( 1)
( ) ( )
( ) ( )
r un m P m P m P m P n m P m P m P m P
s n m P m P m P m P n m P m P m P m P
A C
B D
          
          

         
         
1 1 2 2 3 3 4 4
1 2 3 4
1 2 43
1 2 3 4
1 2 43
1
1 1 2 2 3 3 4 4 !
n m P m P m P m P
g g g g
P P PP
q q q q
P P PP
E E E E x m
F F F F n m P m P m P m P
   
       
      

          
      

3. Finite Triple Integral Representation For The Polynomial Set Tn(x1,x2,x3,x4)
DOI: 10.9790/5728-11610914 www.iosrjournals.org 11 | Page
     
43
31 2 41
1
2 2
1 1 1
3 41 2 3 4 3
1 2 3 42
1
! ! ! ! ( ) ( ) ( )
PPmP P mP
P
m P
P P P
x x a b c d d d
P x P P P a b c
          

1 1 2 21 1
31 2
1 1 1
1 2 3
1 1 1 1 1 1
0 0 0
0 0 0
n m P m Pn n m P
mm m
a P b P c P
P P P
      
    
            
  
         
 
 
1 1 2 2 3 3
4
1 1 2 2 3 3 4 4
4 1 1 2 2 3 3 4 4
( 1) ( 1) ( 1)
0 ( 1) ( 1) ( 1)
( )
( )
n m P m P m P
m
r n m P m P m P m P
sP n m P m P m P m P
A
B
   
 
 
      
       
 
         
         
1 2 3 41 1 2 2 3 3 4 4 1 2 43
1 2 3 41 1 2 2 3 3 4 4 1 2 43
v
( )
( )
u g g g gn m P m P m P m P P P PP
q q q qn m P m P m P m P P P PP
C E E E E
D F F F F
   
   
               

               
       
       
3 41 31 2 4
1
2
2
1 1 1
3 41 2 3 3
1 2 3 42
1
! ! ! !
P PP mP P m
P
Pm
P P P
x x a b c
P x P P P a b c
       

 
 
1 1 2 2 3 3 4 4
1
1 1 2 2 3 3 4 4 !
n m P m P m P m P
m
x
d d d
n m P m P m P m P
   

   
   
1 1 2 21 1 1 1 2 2 3 3
31 2 4
1 2 3 40 0 0 0
n m P m Pn n m P n m P m P m P
mm m m
P P P P
          
      
       
   
    
   
   
1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4
1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4
( 1) ( 1) ( 1)
v( 1) ( 1) ( 1)
( ) ( )
( ) ( )
r un m P m P m P m P n m P m P m P m P
s n m P m P m P m P n m P m P m P m P
A C
B D
          
          

         
         
1 1 2 2 3 3 4 4
1 2 3 4
1 2 43
1 2 3 4
1 2 43
1
1 1 2 2 3 3 4 4 !
m P m P m P m P
m
g g g g
P P PP
q q q q
P P PP
E E E E x
F F F F n m P m P m P m P
  
       
      

          
      
       
     
431 32 4
1
2 2
1 1 1
1 3 42 3 4 3
1 2 3 42
1
! ! ! !
PPP mP m
P
m P
P P P
x x a b c
P x P P P a b c
       

     
 
1 1 1
11 3
a P b P c P
a b c P
     

    

4. Finite Triple Integral Representation For The Polynomial Set Tn(x1,x2,x3,x4)
DOI: 10.9790/5728-11610914 www.iosrjournals.org 12 | Page
     
 
 
 
1 1 2 2 3 3 4 4
1 2 3 4 1 1 2 2 3 3 4 4
( 1) ( 1) ( 1)
, , , 0 ( 1) ( 1) ( 1)
1 ( )
1 1 ( )
s m P m P m P m P
rP P P P m P m P m P m P
B na b c
a b c A n

     
      
   

     

      
   
1 1 1 2 2
1 1 2 2
( 1) ( 1)
1 2
1 1 1 2 2
1 1
! !
P m r s u v P m r s u v r s P
m P m P
m m
x P x x P
         
    

 
… (2.2)
The single terminating factor   1 1 2 2 3 3 4 4m P m P m P m P
n   
 makes all
summation in (2.2) runs upto.
     
 
 1 2 3 4, , ,
1
n
a b c
T x x x x
a b c
  

   
On using [16], hence the theorem.
1 1 1 1 1 1 1
0 0 0
n x y l m m n
x y z z dx dy dz
      
  
     
 1
l m n
l m n
  

   
Where 1x y z  
3 Particular Cases
I. On taking 10    r s u v q , 1 11       g m m ; 1 2 11 , 1     E
and y for x1 in (2.1), we achieve
 2
nA y  
     
1 1 1 1 1 1
0 0 0
1
!
n
a b ca b c y
a b c n
       
   
     
2,1 , 1; 1
, , ;
n a b c
F d d d
a b c y
      
    
 
(ii)If we set 0   r s u v; 1 1 1 11          g q m m ; 1 1( ), ( ) p qE a F b
and 1
1
x
x
  in (2.1) we have
 1 1 , ;F n b x
 
     
1 1 1 1 1 1
0 0 0
1 a b ca b c
a b c
       
   
     

5. Finite Triple Integral Representation For The Polynomial Set Tn(x1,x2,x3,x4)
DOI: 10.9790/5728-11610914 www.iosrjournals.org 13 | Page
 
 
, , 1;
, , , ;
p
q
n a a b c
F x d d d
b a b c
    
    
 
 
(iii) On setting 1 10r s v g q     ; 1 11       u m m , 1 11, 1C     
and
1
x
y
 in (2.1) we have
 
 nA y
      
     
1 1 1 1 1 1
0 0 0
1 1 1
!
n
a b cn
a b c
a b c n
          
   
     
, 1;
, , , ;
n a b c
F y d d d
n a b c
    
       
(iv) If we set 10r s u g    ; 1 11      v q m m ;
1 1
1
1 , 1 ,
2
D F          and 1
1
1
x
x
x



, in (2.1), we get
 ( , )
nP x       
     
1 1 1 1 1 1
0 0 0
1 1 1
! 2
n
a b cn
n
a b c x
a b c n
          
   
     
, , 1; 1
1 , , , ; 1
n n a b c x
F d d d
a b c x
       
       
(v) On taking 10r s u g    ; 1 11      v q m m ; 1
1
2
    ;
1 1 ,D    1 1F   and writing
1
1
x
x


for x1, in (2.1), we get
 
 ,
nP x
       
     
1 1 1 1 1 1
0 0 0
1 1 1
! 2
a b cn
n
a b c x
a b c n

         
   
     
, ; 1; 1
1 , , , ; 1
n n a b c x
F d d d
a b c x
      
       
(vi) On making the substitution 10r s u g    ; 1 11      v q m m ; 1
1
2
    ;
1 11 , 1D F      and writing
1
1
x
x


for x1 in (2.1), we get
 
 ,
nP x
       
     
1 1 1 1 1 1
0 0 0
1 1 1
! 2
n
a b cn
n
a b c x
a b c n
          
   
     
, , 1; 1
1 , , , ; 1
n n a b c x
F d d d
a b c x
       
       

6. Finite Triple Integral Representation For The Polynomial Set Tn(x1,x2,x3,x4)
DOI: 10.9790/5728-11610914 www.iosrjournals.org 14 | Page
(vii) If we take 1 10r s v g q     ; 1 11u m m          ; 1c   and
1
x
for x, in (2.1), we get
 nF x    
     
1 1 1 1 1 1
0 0 0
1
!
a b cn
a b c
a b c n
        
   
     
, 1;
1 , , , ;
n a b c
F x d d d
n a b c
    
       
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