Weyl Fractional Derivative of Multivariable Polynomials and I- Function

IJSRR, 8(2) April. – June., 2019 Page 138

Research article Available online www.ijsrr.org

ISSN: 2279–0543

International Journal of Scientific Research and Reviews

Weyl Fractional Derivative of Multivariable Polynomials and

I- Function

Neelam Pandey

1

, Harish Kumar Mishra

*2

and Kshamapati Tripathi

3

1_{Department of Mathematics, Govt. Model Science College , Rewa (M.P.), India} 2

S.V. M. Science and Technology, Lalganj, Pratapgarh, India 3

Department of Mathematical Sciences, A.P.S. University,Rewa (M.P.), India 2

Email: [email protected]

ABSTRACT

In this research work, we establish a theorem on Weyl fractional derivative of the product of

multivariable polynomials and I– function in the literature of special functions. The results are obtain in the

compact form containing the multivariable polynomials. Some special cases of our theorem have been

discussed.

KEYWORDS-

Weyl fractional derivative operator, multivariable polynomials and I-function.

*Corresponding author

Harish Kumar Mishra

S.V. M. Science and Technology,

Lalganj, Pratapgarh, India

IJSRR, 8(2) April. – June., 2019 Page 139

INTRODUCTION

Recently, for modeling of relevant systems in various fields of sciences and engineering, such

as fluid flow, diffusion, relaxation, oscillation, anomalous diffusion, polymer physics, chemical

physics, propagation of seismic waves, etc. see, Glöckle and Nonnenmacher1 , Mainardi2, 3, Miller

and Ross4, Kilbas et al.5 and others.

The I-function of the one variable is defined by Saxena6 and we will represent here in the

following manner:

_

     



 

 

, , ,

; , I ] [ ,

; , I z] [

I m n z

r q p z n m

r q

pi i i i 

  

    

 

qi m ji ji m j j

pi n ji ji n j j n

m r q p

f b f

b

e a e

a z

i i

, 1 , ,

1

, 1 , ,

1 ,

; ,

) ( ; ) , (

) ( ; ) , (

I (1.1)

ds z s i

s L



 ( )

2 1



 , (1.2)

wherei (1),z(0) is a complex variable and (1.2) zs



s



z i z





arg | | log

exp 

 .In which log

|z| represent the natural logarithm of |z| and arg |z| is not necessarily the principle value. An empty

product is interpreted as unity. Also,











   

 

   

 

  

 

   

 

r

i

p

n j

ji ji q

m j

ji ji

n

j

j j m

j

j j

i i

s e a s f b

s e a s

f b s

1 1 1

1 1

) ( ) 1

(

) 1

( ) ( )

(

 (1.3)

m, n, pi and

q

_i



i



(

1

,



r

)

are non-negative integers satisfying

0



n



p

i

,



1, ,



, ,( 1, , ; 1, ) ;

0mq_i i  r e_ji j   p_i i  r and f_ji,



j1,,q_i;i1,r



are assumed

to be positive quantities for standardization purpose. Also a_ji,(j1,,p_i;

i



1

,



r

)

and

; , , 1 (

, _i

ji j q

b  

i



1

,



,

r

)

are complex numbers such that none of the points.



(  )



, 1,, ; 0,1,2,,

 b v f h mv

S _{n h}

(1.4)

Which are the poles of(bh fhs),h1,,m and the points



(





1

)





1

,



,

;



0

,

1

,

2

,



,



a

n

e

l

n



S

_{l l}

(1.5)

Which poles are of (1al els) coincide with one another, i.e. with

) 1 (

)

( n   n l 

l b v b a

IJSRR, 8(2) April. – June., 2019 Page 140 forv,0,1,2,;h1,,m;l1,,n.

Further, the contour L runs from



i

_ to



i

_. Such that the poles of



(

b

_n



s

)

,

h



1

,



,

m

;

lie to the right of L and the poles (1al els), l



1

,



,

n

lie to the left of L. The integral

converges, if |argz| , 0, 0 2

1

 

 B B A , where





 

 

i

i q

j ji p

j

ji f

e A

1 1

and (1.7)

) ,..., 1 ( , 1 1

1 1

r i f f e e B

i

i q

m j

ji m

j j p

n j

ji n

j

j    



_

_

_

_

     

(1.8)

Let A denote a class of good functions. By good function f, we mean Miller7 a function which is

everywhere differentiable any number of times and if it is all of its derivatives are 0(x), for all 

as x in increases without limit. We define the Weyl fractional derivatives of a function g(z) as

follows:-

LetgA, then forq0,

. ) ( ) ( ) (

) 1 ( ) (

1 du u g z u q z

g W

q

z q q

z

  







 



 (1.9)

For q0

)) ( ( )

( W g z

dz d z g

W n z q n

n q

z

 

  , (1.10)

Where n being positive integer, such that

n



q

.

The general class of multivariable polynomials is defined by Srivastava and Garg8.





! ! ) , , ; L ( L)

( ,

,

1 1 1

L

0 , , 1

, , L

1 1

1

1

1 1 1

r k r k r k

h k h

k k

k h k h r

h h

k x k x k k A x

x S

r r

r

r

r r

r _{  }





 



  



  

 , (1.11)

Where h1,...,hr are positive integers and the co-efficient A(L; k1,...,kr), (L; h1 N; i =

1,...,r) are arbitrary constant, real or complex.

Evidently the caser1 of the polynomials(1.11).

IJSRR, 8(2) April. – June., 2019 Page 141

 



L (0,1,2, )



! L) (

, L ]

, [

0

L   





_



N x

A k x

S k

k h

L

k

hk

h _{, (1.12)}

where h is arbitrary positive integers and the co-efficientAL,k(L,k0) are arbitrary constant, real or

complex.

2. MATHEMATICAL PRE-REQUISITES - To establish the main result, we need the following

integral of the H-function by SaigÖ10.





dt

f b f

b

e a e

a x t zt x

t t

i i i

i

q m ji ji m j j

p n ji ji n j j n

m r q p

x _

 

   

 



  

 



,, ; ₁1_,, 1₁,_,

1 1

) , ( ; ) , (

) , ( ; ) , ( )

(   



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

(

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

, 1 ,

1 , 1 ,

1 1

, 1

; 1 , 2 1

   

   

  

 

 

 

 

    

i i

i i

q m ji ji m j j

p n ji ji n j j n

m r q p

f b f b

e a e a zx

x

   

  



  

 _(2.1)

Where

(i)



,



are complex numbers and, are positive real numbers,

(ii) argz 1₂A, A defined as _, 1

1





 

 

i

i q

j ji p

j

ji f

e

A

(iii) _

   

   

   

   

             

 

  

j j

f b m j Re 1

min , 1

Re min

 



 _{max Re ,max1 Re} 1 _.

   

   

   

   

      

 

        

j j e a N j v



3.

WEYL

FRACTIONAL

DERIVATIVE

OF

MULTIVARIABLE

POLYNOMIALS AND I- FUNCTION-

THEOREM. Let

m

,

n

,

p

_i and

q

_i be non-negative integers such that

0



n



p

_i,

0



m



q

_i and

0 1 1

1 1

 





_

_

_



     

i

i q

m j

ji m

j j p

n j

ji n

j

j e f f

e together with the set of conditions (i) – (iii) given with equation

(2.1). Then, for all value of q,







_x

_z

_x

_S

r

_c

_x

_c

_x

r

W

q _Lh h _r

z

 

 

,

,

)

(

1 1

1 , , 1

1  

_



IJSRR, 8(2) April. – June., 2019 Page 142                   i i i i q m ji ji m j j p n ji ji n j j n m r q

p _{b f b f}

e a e a z x yx , 1 , 1 , 1 , 1 , ; , _{( , ) ;( , )} ) , ( ; ) , ( ) (   ! ! ) , , ; ( ) ( ) ( ) 1 ( 1 1 1 0 , , 2 1 1 1 1 1 1 1 1 r k r k r k h k h L k h k h k k k q q k c k c k k L A L z q r r r r r r r

i ii  

   _{ }           



                                          i i i i q m ji ji m j j r

i i i

r

i i i p n ji ji n j j n m r q p f b f b k q k e a e a q yz , 1 , 1 1 1 , 1 , 1 1 , 1 ; . ) , ( ; ) , ( ), , 2 ( ) , 1 ( , ) , ( ; ) , ( ), , 2 ( 1 2            

. (3.1)

PROOF - Taking left hand side of equation (3.1) and using equation (1.11), we get

                 



 ! ! ) , , ; ( ) ( ) ( 1 1 1 0 , , 1 1 1 1 1 1 1 1 1 r k r k r k h k h L k h k h k k k q z k c k c k k L A L x z x W r r r r r r r

i ii  

                        i i i i q m ji ji m j j p n ji ji n j j n m r q p f b f b e a e a z x yx , 1 , 1 , 1 , 1 , ; , ) , ( ; ) , ( ) , ( ; ) , ( ) ( 

 _{, (3.2)}

Now using equation (1.9) and definition of I– function, easily we can find the proof of

equation (3.1).

For q ≥ 0 invoking the definition (1.10) the relation (3.2) further reduces to

! ... ! ) ,..., ; ( ) ( 1 1 1 1 1 1 1 1 1 1 1 ... ... 0 ,..., r k k r k h k h L k h k h k k k c k c k k L A L r r r r r r        

_

{ . ) ( ) 1

( 1 2

1         _     i i r i k r q r r r q z dz d q r     } ) , ( ; ) , ( ), , 2 ( ) , 1 ( , ) , ( ; ) , ( ), , 2 ( , 1 , 1 1 1 , 1 , 1 1 , 1 ; . ₁ 2                                 i i i i q m ji ji m j j r

i i i

r

i i i p n ji ji n j j n m r q p f b f b k q k e a e a q r yz            

In replacing of (q - r) by q, we may obtain again

! ! ) , , ; ( ) ( ) ( ) 1 ( 1 1 1 0 , , 2 1 1 1 1 1 1 1 1 r k r k r k h k h L k h k h k k k q q k c k c k k L A L z q r r r r r r r

i i i  

   _{ }           



                                          i i i i q m ji ji m j j r

i i i

r

IJSRR, 8(2) April. – June., 2019 Page 143 4. SPECIAL CASE - If we put r=1 in the general call of multivariable polynomials given by

Srivastava and Garg8 reduces to the polynomials given by Srivastava(1972) and I- function reduces

into Fax , s H – function as follows :

 



                         q p k h q z q

z _{b f}

e a z x yx q p n m H x S x z x W W , 1 1 1 , 1 1 1 L 1 1 ) , ( ) , ( ) ( , , ) (       k k L k hk k q q A k z

q L,

, 0 2 1 ! L) ( ) ( ) 1 (



                                      q j j p j j n m q p f b k q k e a q yz H , 1 , 1 1 , 1 1 ,. 2 ) , ( ), , 2 ( ) , 1 ( , ) , ( ), , 2 (         

 _{. (4.1)}

Replacing  by – equation (4.1) correspond to the following result according as μ > , μ <  and

μ = , i.e. for μ > 

 



                          q p k h q z q z f b e a z x yx q p n m H x S x z x W W , 1 1 1 , 1 1 1 L 1 1 ) , ( ) , ( ) ( , , ) (       k h L k hk k q q A k z

q L,

, 0 2 1 ! L) ( ) ( ) 1 (



                                    q j j p j j n m q p f b q k q k e a yz H , 1 , 1 , 2 2 ,. 1 ) , ( ), , 1 ( ), , 2 ( ) , 1 ( , ) , (           (4.2)

For μ < 

 



                          q p k h q z q

z _{b f}

e a z x yx q p n m H x S x z x W W , 1 1 1 , 1 1 1 L 1 1 ) , ( ) , ( ) ( , , ) (                                         



q j j p j j n m q p k h L k hk k q q f b q k e a k q yz H A k z

q 1,

, 1 1 , 1 1 ,. 2 , L , 0 2 1 ) , ( ), , 1 ( ) , 1 ( , ) , ( ), , 1 ( ! L) ( ) ( ) 1 (              (4.3)

and 

 



                          q p k h q z q

z _{b f}

e a z x yx q p n m H x S x z x W W , 1 1 1 , 1 1 1 L 1 1 ) , ( ) , ( ) ( , , ) (                                      



q j j p j j n m q p k h L k hk k q q f b q k e a y H A k z q k q , 1 , 1 , 1 1 ,. 1 , L , 0 2 1 ) , ( ), , 1 ( ) , 1 ( , ) , ( ! L) ( ) ( ) 2 ( ) 1 (         (4.4)

Finally writing



instead of



, equation (4.1) yields the following results according as,



 and respectively.

IJSRR, 8(2) April. – June., 2019 Page 144



 

k h q

zW x z x SL x

1 1

)

( 

  

 



   

   



 

q p f b

e a z x yx q p

n m H

, 1 1 1

, 1 1 1

) , (

) , ( ) ( ,

, _{ }

 

   

   

        

 

 

   

  

     



( L)_! ( ₍1,_{, )} )_,,₍(2 _{, )}, ),( , )

) (

) 1 (

, 1

, 1 2

, 1 ,. 2 , L ,

0 2 1

 

  

 



  

 

k f

b

e a q

k q

yz H

A k z

q j j q

p j j n

m q p k h

L

k hk k

q q

(4.5)

for

 



    

    

   

   

 

  

  

q p k

h q

z q z

f b

e a z x yx q p

n m H x S x z x W W

, 1 1 1

, 1 1 1 L

1 1

) , (

) , ( ) ( ,

, )

(   



   

   

 

   

  

 

 

    

      



( L)_{! (2} (2 _, , ),_),(₍ ,_, )_{) ,( , )}

) (

) 1 (

, 1 , 1 1

, 1

1 ,. ,

L ]

, [

0 2 1

  

  

 

  

 

k f

b k

q

e a q yz

H A k z

q j j q

p j j n

m q p k h

L

k hk k

q q

(4.6)

and for

 



    

    

   

   

 

  

  

q p k

h q

z q z

f b

e a z x yx q p

n m H x S x z x W W

, 1 1 1

, 1 1 1 L

1 1

) , (

) , ( ) ( ,

, )

(   



k h

L

k

hk k

q q

A k z

q q k

, L ]

, [

0 2 1

! L) ( )

(

) 1 ( ) 2

(



      

 



     

  



 

   

   

    

  



   

 

q j j p j j n

m q p

f b k

q

k e

a q yz

H

, 1 ,

1 1

, 1

1 ,. 2

) , ( ), , 2

(

) , 1 ( , ) , ( ), , 2 (

   

  





 _{. (4.7)}

DISCUSSION

We have obtained the results namely theorem and special cases which satisfied all the conditions

mention in the statement.

REFERENCES

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6434.

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differential equations Wiley, New York 1993.

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differential equations.AMADE2003, Cambridge Scientific, Cambridge 2006; 117-134.

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