Some Finite Integrals Of Generalized Polynomial Sets And The Multivariable Aleph -Function With Applications

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Some Finite Integrals Of Generalized Polynomial

Sets And The Multivariable Aleph



-Function With

Applications

Yashwant Singh

Abstract: In the present paper, the author will evaluate three new finite integrals with the product of generalized polynomial sets and the multivariable Aleph (



)-function. These integrals are unified in nature as a key formula from which we can derive its particular cases as integrals involving a large number of simpler special functions and polynomials. At the end, we give applications of our main findings by inter-connecting them with Riemann-Liouville type of fractional integral operator.

Key words: Multivariable Aleph (



)-function, Multivariable

H

-function, Generalized Polynomial, Riemann-Liouville fractional derivative.(2010 Mathematical Subject Classification: 33C99)

————————————————————

1. INTRODUCTION

The multivariable Aleph-function is defined by Ayant [2] and represented by means of the multiple Mellin-Barnes type integrals in the following manner:





1 1

(1) ( )

(1) (1) (1) ( ) ( ) ( )

1

0, : , ;..., ,

1

,...,

_, _{, , :} _, _, _; _;...; _, _, _;

r r

r i i i _i _i _i _ir _ir _ir

n m n m n

r _{p q} _{R p} _q _R _p _q _R

r

z

  







 

_















(1) ( )





(1) ( )





(1) (1)



1, 1, 1, 1

(1) ( ) (1) (1)

1, 1, 1

; ,..., , ; ,..., : , ,

..., ; ,..., : , ,

r r

j j j i ji ji ji j j

n n pi n

r

i ji ji ji _q j j _m i

a a c

b d

         



     

     

          _ _

 









(1) (1) ( ) ( )

(1) (1) (1) _1, (

1,

1 (1)

(1) (1) ( ) ( ) ( ) ( )

(1) (1) (1) _1, ( ) ( ) ( )

1, 1,

1 (1) ( )

, ;...; ,

, ;...; , ,

r r

j j r

i ji ji n p nr i i

r r r r

j j r r r

i ji ji m q_i mr i ji ji mr q_ir

c c

d d d

   

    



 

 _ _  _ _  _ _  _ _      

 

 _ _  _ _ _ _   _ _  _ _ _ _          

   

( ) ( )

) ( ) ( )

1, ( ) ,

r r r r ji ji _n _p

r _ir

c 



 _ _   _ _     

 







=

1

1 1

1

1 ...

( ,...,

)

( )

...

(2

)

k

r

s

r k k k r

r

k

L L

s

s z ds

ds

w







 





(1.1)

Where

w

 

1

( )

1 1

1

( ) ( )

1 1 1 1 1

1 ( ,...,

)

1

i i

n r

k

j j k

k j

r _R _p _r _q _r

k k

i ji ji k ji ji k

i j n k j k

a

s

a

s

b

s











 

     





 

_



_









_



_



_{ }



_











































(1.2)



 





 



( ) ( ) ( )

( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( )

1 1

( ) ( ) ( ) ( ) ( )

1 1

1 ( )

1

k k

k _ik _ik

k k k k

k

k k

m n

k k k k

j j k j j k

j j

k k _R q p

k k k k k

i _ji _ji k _ji _ji k

j m j n

i

d

s

c

s

d

s

c

s













 

   





 









 





















 



(1.3)

The real numbers



_iare positive for

i



1,..., ;

R



_i( )k are

positive for

i

( )k



1,...,

R

( )k .

The conditions for absolute convergence of multiple Mellin-Barnes type contour (1.1) are given as:

( )

1 | arg

|

2

k

k i

z



A



(1.4)

Where

( ) ( ) ( ) ( ) ( )

1 1 1 1

i i k

p q n

n

k k k k k

i j i ji i ji j

j j n j j

A













    

















( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

1 1 1

0

k k

i k i

k k k k

k k

p _m q

k k k

j

i ji i ji

j n j j m











    















(1.5)

With

k



1,..., ;

r i



1,..., ;

R i

( )k



1,...,

R

( )k

The complex numbers

z

_iare not zero. Throughout this document, we assume the existance and absolute

convergence conditions of the multivariable Aleph-function. We may establish the asymptotic expansion in the following convenient form:







1







1

,...,

|

1

| ,...,|

|

, max |

1

|,...,|

|

0

r

r r r

z

o z



z



z







,







1







1

,...,

|

1

| ,...,|

|

, min |

1

|,...,|

|

r

r r r

z

o z



z



z





 

__________________________

 Government College, Kaladera, Jaipur, Rajasthan, INDIA

(2)

3142

Where

( )

1,..., ;

min Re

,

1,...,

k j

k k k

j

d

k

r



j

m

















_



_







_

_







and

( )

1 max Re

,

1,...,

k j

k k k

j

c

j

n





















_



_







_

_







For convenience, we will use the following notations in this section:

1

, ;...;

1 r

,

r

V



m n

(1.6)

(1) (1) (1) ( ) ( ) ( )

(1) ( )

,

;

;...,

r

,

r

,

r

;

r

i i i i i i

W



p

q



R

p

q



R

(1.7)













1

(1) ( ) (1) ( ) (1) (1)

1, 1, 1,

;

,...,

,

;

,...,

:

,

i

r r

j j j _n i ji ji ji _n _p j j _n

A

a





a



c







 









_

_

_

_

_

_









,





(1) (1) ( ) ( ) ( ) ( )

(1) (1) (1) _1, ( ) ( ) ( )

1, 1,

1 (1) ( )

, ; ...; jr, jr r rr , rr

i ji ji n p_i nr i ji ji nr p_ir

c c c

    

 

  _ _ 

 _ _  _ _ _ _   _ _  _ _ _ _ 

   

   

(1.8)









1

(1) ( ) (1) (1)

1, 1,

;

,...,

:

,

i

r

i ji ji ji _q j j _m

B





_



b



 

_{ }

_

d





_







(1) (1)



( ) ( )



( ) ( )

(1) (1) (1) _1, ( ) ( ) ( )

1, 1,

1 (1) ( )

, ; ...; r , r r , r

j j r r r

i ji ji m q_i mr i ji ji mr q_ir

d d d

    

 

  _ _ 

 _ _  _ _ _ _   _ _  _ _ _ _ 

   

   

(1.9)

The contracted form concerning the multivariable Aleph-function is given as:





1

0, : 1

,...,

i, , ; :i i

r

A z n V

r p q R W _z

B

z

_







 



_



_





(1.10)

The multivariable

H

-function of several complex variables occurring in this paper by Srivastava and Panda [10] is reported by means of the multiple Mellin-Barnes integrals in the following form:







 









1 ( )

 

1 1





( ) ( )



1,

1, 1,

1 1

1 1 1 1

( ) ( ) ( )

1 1 1

1,

1, 1 1,

; ,..., : , ;...; ,

0, ; , ;..., ,

1 , ; , ;..., ,

; ,..., : , ;...; ,

,...,

r r r

j j j _p j j _p j j pr r r

r r

r r r r

j j j j j _q j j

q qr

a c c

z n m n m n

r p q p q p q

z

b d d

H z

z

H

       



















=

1

1 1 1 1 1

1 ...

( )... ( ) ( ,...,

)

...

(2

)

r

r r r r r

r

L L

z

z d

d

w

 

 

   





 

(1.11) Where

w

 

1 















( ) ( ) ( ) ( )

1 1

( ) ( ) ( ) ( )

1 1

1 ( )

1

i i

m n

i i i i

j j i j j i

j j

i i q p

i i i i

j j i j j i

j m j n

d

c

d

c

 

 

 

 

 

 

   





 





 









1,...,

i

r

 

(1.12)

( )

1 1

1

( ) ( )

1 1

1 ( ,...,

)

1

n r

i

j j i

i j

r p r q r

i i

j j i j j i

i i

j n j

a

b

 

 



 

 

 

  





 

_



_

















_



_

  

_

_

















(1.13)

The generalized polynomials (multivariable) is defined by Srivastava [9] in the following manner:





1 1

1 1 1

1

1 ,...,

,..., 1

0 0 1

(

)

(

)

,...,

...

(

)!

(

)!

R R

R R R

R

Q Q

P P

P R P

P P

Q Q R

R

Q

S

x

 



             



 





1

,

1

;...;

,

1

,...,

R

R R R

A Q



Q



x



x

 (1.14)

Where

Q j

_j

(



1,..., )

R

are non-zero arbitrary positive intgers. The coefficients

A Q



₁

,



₁

;...;

Q

_R

,



_R



being arbitrary constants, real or complex.

If we take

R



1

in the equation (1.14) and denote



1

,

1



A Q



thus obtained by

A Q



_i

,



_i



, we arrive at the well known general class of polynomials

S

_QP

[ ]

x

introduced by Srivastava [8].

The general polynomial set defined by Raizada [6] in the following Rodrigues type formula:

, , , , ( )/

[ , , , , , , , ]

[ ]

(

) (1

)

r

n n

S

  

x r s q A B k l



S

  

x



Ax



B







x

 

T

_{k l}m n_,

_



(

Ax



B

)

qn

(1





x

r

)

( / )sn



_

(1.15)

Where the differential operator

T

_{k l}_, is defined by

,

l k l

d

T

x

k

x

dx







_



_





(1.16)

Though the polynomial set

S

n, ,

[ ]

x

  

can also be derived from the general sequence of function defined by Agarwal and Choubey [1], it unifies and extends a number of classical polynomials studied by various research workers such as Gould and Hooper [3], Gradshtiyn and Ryshk [4], Krall and Frink [5], Singh and Srivastava [7] etc. Moreover, it can be expressed in the following series form as:

, , ( ) ( )

[ , , , , , , , ]

qn l m n

(1

r sn

)

m n

n

S

  

x r s q A B k l



B x







x

l



0 0 0 0

( 1) (

) (

) ( ) (

)

! ! !(1

)

p

m n m n

t i i

p t i i

p

qn

p

i

 

  



 

     



 



1

p r

r

p m n

i

k

rt

x

Ax

sn

l

x

B







_









 





_

 













 



_







 



_

_





(1.17)

(3)

, ,0 ( ) ( )

0 0

(

)

[ , , , ,1, 0, , ]

! !

p m n

qn l m n m n t

n

p t

p

S

x r s q

k l

x

l

p t

       







(

r

)

p

m n

qn

k

rt

x

l



_



 













(1.18)

2. Main Integrals:

In this section, we will derive the following integrals: First Integral:

1 1 , ,0

(

)

(

)

(

)

; , ,1, 0, , ,

v b

n a

x a

b

x

x a

b

x

x c

S

y

r q

m k l

x b

x c



      







 









_

_

_{ }

_

_





 













1 1

,...,

,..., 1

,...,

R R

h k h k

P P

Q Q R

x a

b

x

x a

b

x

S

y

x c



_

_

_{ }

_

_

_

_

_{ }

_

_





_

_{ }

_

_

_{ }

_







 





 











1 1

1

,...,

r r

g w g w

r

x a

b

x

x a

b

x

z

dx

x c



_

_

_{ }

_

_

_

_

_{ }

_

_







_

 

_





_

 

_







 





 











=

(

b a



)

    ( v qn l m n)[ (  ) rp] 1

(

b c



)

  [qn l m n(  ) rp]

[ ( ) ] ( )

(

a c



)

  v qn l m n  rp

l

m n

g

qn l m n  rp

1

0 0

(

)

( ,....,

)

! !

p p

m n

f R

p f m n

p

_{qn k}

_rf

L y

y

p f

l







  







 













1 1

1 2

3

, ,

0, 2:

2, 1, ; : 1

,...,

,

r r

i i i

g w g w

F F A n V

p q R W r B F

b a

z

b c

   



_

_

_{ }

_

_

_

_

_{ }

_

_







_

_{ }

_

_

_{ }

_







 





 











(2.1) Where

 

1

' '

1

' '

1

' _,

1 '

0 0 ' 1 '

( ,...,

)

...

!

R

j j

R

j j

R

Q Q

h k

P P R _j

p

r j

j j

q

b a

L y

y

b c



 



           

  





_

_

_

_

_{ }

_

_

_

_

_





_





_

 

_









 













  

A Q

[

₁

,



₁

;...;

Q

_R

,



_R

]

(2.2)

1

(1

[

(

)

]

1 1

...

R R

;

1

,...,

r

)

F

  

 

qn l m n



 

rp



h



 

h



g

(2.3)

1

(1

[

(

)

]

1 1

...

R R

;

1

,...,

r

)

F

  



v qn l m n



 

rp



k



 

k



w

(2.4)

3

(1

(

)[

(

)

] (

1 1

)

1

...

F

   

 





v qn l m n



 

rp



h



k





(

h

_R



k

_R

)



_R

;

g

₁



w

₁

,...,

g

_r



w

_r

)

(2.5)

Provided that

' ' '

0 {1,..., };

,

0 ' (1,..., }

( ' 1,..., )

i i j j j

g w

  

i

r h k

  

j

R P j



R

is an arbitrary positive integer and the coefficients

1 1

[

,

;...;

_R

,

_R

]

A Q



Q



being an arbitrary constant, real and complex.

( ) ( )

1 1

Re

0; Re

0

i i

r r

j j

i i

i j i j

d







 





























_



_







_



_





_

_





_

_















( )

0; | arg

|

1

( )

,

.

2

k k

i k i

A



z



A



c

 

a

b

(2.6)

Second Integral





( , ) , ,0

0

(

)

(1

)

; , ,1, 0, , ,

t

v n

t



x

 

x P

 





x S

_ 



_

y t



x



x r q

m k l



_







1





1 1

1 ,...,

,..., 1

,...,

R

R R

R

h h

P P k k

Q Q R

S



_

y t



x

y

t



x



_

1 1

1

(

)

,...,

(

)

r r

g w g w

r

z t

x

z t

x

dx







_



_

=

( v q)[ l m n( ) rp] 1

t

        

1

0 0

(

)

( ,...,

)

! !

p p

m

f r

p f m

p

q

k

rf

L y

y

p f

l





_

_



  



_

_

_{ }

_











1 0

( ,...,

) (

1) (

) (

1)

2

s n

R s s

s

t

D y

y



n



n











  



  

_

_







1 1 1

2 ,

0, 2:

1, 1, ; : 1

,...,

,

r r i i i

g w g w D A

n V

p q  R W

z t

r B D

  

 







_

_

(2.7)

Where

 

1 1

' ' ' ' '

1

' _, ₍ ₎

1 ' 1 1

0 0 ' 1 '

( ,...,

)

...

2 [

,

;...;

,

]

!

R R

j j j j j

R

Q Q

P P R _j

p h k

r j R R

j j

q

D y

y



y



A Q

Q

 



           

   







_



_





  

(2.8)

1

[

(

)

1 1

...,

r r

;

1

,...,

r

]

D

  



v q





l m n

 

rp



k





k



w

(2.9)



2

1 (

)(

(

)

D

   

 





v q





l m n

 

rp





1 1 1 1 1

(

h



k

)





..., (

h

_r



k

_r

)



_r

;

g



w

,...,

g

_r



w

_r (2.10)

And

P

_n( , ) 

( )

x

is the well known Jacobi polynomials.

The integral (2.7) converges under the conditions mentioned in (2.1).

Third Integral





( , ) , ,0

0

1 (

)

(1

)

; , ,1, 0, , ,

2

t

v n

x

t

x

P

x S

y t

x

x r q

m k l



      





_







_

_



_

_



_



_











1





1 1

1 ,...,

,..., 1

,...,

R

R R

R

h h

P P k k

Q Q R

S



_

y t



x

y

t



x



_

1 1

1

(

)

,...,

(

)

r r

g w g w

r

z t

x

z t

x

ds





(4)

3144

=

( )

1

0 0 0

(

)

( ,...,

)

! !

p p

m n

f U m n g l m n rp

r

p f s m

p

q

k

rf

t l

y

D y

y

p f

l

 









    

   







 













(

1) (

)

2 ! !

s

t

s





 





 

 













1 1 1

2 ,

0, 2:

1, 1, ; : 1

,...,

,

r r i i i

g w g w D A

n V

p q  R W

z t

r B D

  

 







_

_

(2.11)

Where

D y

( ,...,

₁

y

_R

)

is same as given in (2.8).

1

[

(

)

1 1

...,

r r

;

1

,...,

r

]

D

  



v q





l m n

 

rp



k





k



w

(2.12)



2

1 (

)(

(

)

D

    

 





v q





l m n

 

rp





1 1 1 1 1

(

h



k

)





..., (

h

_r



k

_r

)



_r

;

g



w

,...,

g

_r



w

_r (2.13)

(

)[

(

)

] 1

U

  

 





v q





l m



n



rp



. (2.14)

Proof of (2.1): We first express the generalized polynomials occuring on the L.H.S. in the series form given by (1.14), the generalized polynomial set by equation (1.17) and the multivariable Aleph-function involved there in terms of Mellin-Barnes contour integral by (1.1). Now we intechange the order of summation and integration (which is

permissible under the conditions stated), so that the L.H.S. of (2.1) (say



) assume the following form :

 

1 1

' ' '

1

' _,

' 1 1

0 0 ' 0 '

...

[

,

;...,

,

]

!

R R

j j j

R

Q Q

P P R _j

P

j R R

j j

R

y

A Q

Q

   



           

  



_







 









  

( )

0 0

(

)

! !

p p

m n

f m n qn l m n rp

p f m n

p

qn

k

rf

l

g

p f

l





    

  







 













1

1 1

1

1 ...

( ,...,

)

(

)

...

(2

)

k

r

r k k k r

r

k

L L

z d

d

w



 



 





 





1 1 1 1

[ ( ) ] 1 [ ( ) ] 1

(

)

(

)

R r R r

j j i i j j i i

j i j i

b qn l m n rp h g v qn l m n rp k w

a

x a

b

x

      

   

             



_



_







1 1

( )[ ( ) ] ( ) ( )

1

(

)

...

R r

j j j i i i

j i

v qn l m n rp h k g w dx

r

x c

d

    



 

       



 







_



_



(2.15)

On evaluating the inner integral of (2.15), we get

( )[ ( ) ] 1 [ ( ) ]

(

b a



)

    v qn l m n  rp

(

b c



)

  qn l m n  rp

[ ( ) ] ( )

(

a c



)

  v qn l m n  rp

l

m n

g

qnm n rp

0 0

(

)

! !

p p

m n

f

p f m n

p

qn

k

rf

p f

l







  







 













1

' '

1

', '

1

'

0 0 ' 1 '

(

)

...

!

R

j j

R

j j

R

Q Q

h k

P P R _j _p

j

j j

q

_{b a}

y

b c



 



           

  







_

_

_

_{ }

_

_



_









_

 

_









 







_



_







  

1

1 1

1 [

,

;...,

,

]

...

(2

)

r

R R r

L L

A Q

Q

w





 



 

(

qn l m n

(

)

rp



h

1 1



...,

h

R



R

;

g

1 1



...

g

r r



]

 



 



 





v qn l m n

(

)

rp



k

1 1



...,

k

R



R

;

w

1 1



...

w

r r



]

 



 



 





1 1 1

1 (

)(

qn l m

(

n

)

rp

(

h

k

)

 

 







_

 



 





1 1 1

..., (

h

R

k

R

)



R

; (

g

w

)



... (

g

r

w

r

)



r

 



 



1

1 1

1

( )... ( ) ( ,...,

1 1

)

1

...

g w

r r r

b a

z

b c



     





_





_



 

_{ }





_





_







_



 





_



1

...,

r

r r

g w

r r

b a

z

d

b c







_







 





_





_

_

 

_{ }

_



_



_





_{ }

(2.16)

Finally, on reinterpreting the Mellin-Barnes contour integral in the R.H.S. of (2.16) in terms of the multivariable Aleph-function given by (1.1), we easily arrive at the desired result.

The integral (2.7) and (2.11) can also prove by similar lines. 3. Special Cases:

In (2.1), (2.7) and (2.11), if we set

(1) ( )

...

r

1;

...

1,

1

r i

i i

R



 





 







, the

Aleph-function reduces to the multivariable H-Aleph-function due to Srivastava and Panda [11] and we get: