International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.

1. International Journal of Computational Engineering Research||Vol, 04||Issue, 2||
Fractional Derivative Associated With the Generalized M-Series and
Multivariable Polynomials
1,
Ashok Singh Shekhawat , 2,Jyoti Shaktawat
1,
Department of Mathematics Arya College of Engineering and Information Technology, Jaipur, Rajasthan
2,
Department of Mathematics Kautilya Institute of Technology and Engineering, Jaipur, Rajasthan
ABSTRACT
The aim of present paper is to derive a fractional derivative of the multivariable H-function of Srivastava
and Panda [9], associated with a general class of multivariable polynomials of Srivastava [6] and the
generalized Lauricella functions of Srivastava and Daoust [11] the generalized M-series. Certain special
cases have also been discussed. The results derived here are of a very general nature and hence
encompass several cases of interest hitherto scattered in the literature.
I.
INTRODUCTION
In this paper the H-function of several complex variables introduced and studied by Srivastava and
Panda [9] is an extension of the multivariable G-function and includes Fox’s H-function, Meijer’s G-function of
one and two variables, the generalized Lauricella functions of Srivastava and Daoust [11], Appell functions etc.
In this note we derive a fractional derivative of H-function of several complex variables of Srivastava and Panda
[9], associated with a general polynomials (multivariable) of Srivastava [6] and the generalized Lauricella
functions of Srivastavaand Daoust [11].Generalized M-series extension of the both Mittag-Laffler function and
generalized hypergeometric functions.
II.
DEFINITIONS AND NOTATIONS
By Oldham and Spanner [4] and Srivastava and Goyal [7] the fractional derivative of a function f(t) of
complex order 
t

1
  1
f(x) dx, Re(    0

 0  t  x)
 

D  f(t)}  
a
t
m
 d
m
D
 f(t)} 0  Re(    m
 m a t
 dt
Where m is positive integer.
The multivariable H-function is defined by Srivastava and Panda [9] in the following manner
0     u' , v' ) ;...; (u
H [z  z   H
1

r
1
 2  i)
where
r
L
i 
(r)
A, C : [B' , D' ] ;...; (B

1
L

1
v
(r)
   
1
(r)
D
z
1


 z
 r

(r)

 a) :  ',..., 
(r)
   b' ) :  ' ] ;...; [b
 c) :  ',..., 
(r)
   d' ) : ' ] ;...; [d
        
r
1
1
r
z
r

1
1
 z

r
r
(r)
(r)


(r)
(r)
d   d  
1


…(2.1)





…(2.2)
r
r
1 .
The general class of multivariable polynomials defined by Srivastava [6] defined as
S
p   p
1
s
q   q
1
s
q
 x  x
1
s
||Issn 2250-3005 ||

1
p

1
k 0
1
q


s
p

k
s
s
0

q 
1
p k
1 1
k 
1
q

s
k

s
p k
s s

||February||2014||
Page 35

2. Fractional Derivative Associated With The…
 A [q  k  q  k  x
1
where
q
1
s
 0 1 2  p
j
j
s
k
1
1
 x
k
…(2.3)
s
s
 0  j  1,..., s) are
non-zero
arbitrary
positive
integer
the
coefficients
A [q  k  q k  being arbitrary constants, real or complex.
1
1
s
s
The following known result of Srivastava and Panda [10]
Lemma. If ( ≥ 0), 0< x < 1, Re (1+p) > 0, Re(q) >  1, i > 0 and i > 0 or i = 0 and | zi | <  , i = 1,2,…,r then
x

 z x 1
 1

F




z x r
r

. F








 1  p  q  2M) (   

M ! (1  p  q  M)
 0
1
r
2
 1  p)

 1
  M, 1 p  q  M ; 
x
 1 p

;


 z  z  F
M
M
1
…(2.4)
where
F
M
E  2 : U' ;...; U
 z  z   F
1
r
p  2 : V' ;...; V
  e) :  ';...;  (r)  1  p    1   r 

(r)
  g) :  ';...;    2  p  q  M     1   r 

(r)
(r)
(r)
x
(r)
   M  1;       v' ) : t' ] ;...; [(v
1
r
(r)
   1      w' ) : x' ] ;...; [(w
r

 t
(r)


z  z 
1
r


…(2.5)
where M ≥ 0,
In this paper, we also use short notations as given
 
 
 1
 1
  F   
F
(r)
P : V' ,..., V




 
 
 t
 r
denote the generalized Lauricella function of several complex variable.
The special case of the fractional derivative of Oldham and Spanier [4] is
E : U' ,..., U

(r)
    1


…(2.7)
Re(     1
      1
The generalized M-series is the extension of the both Mittag-Leffler function and generalized hypergeometric
function.
It represent as following
D
t
t 
…(2.6)
t
 
 
M  c  c  d  d  z)  M  z)
1
p, q



k0
p
1
q
 c   c 
1
k
p
(d   d 
1
k
q
k
k
p, q
z
k
k   
III.
z,     c, Re(    0
…(2.8)
THE MAIN RESULT
Our main result of this paper is the fractional derivative formula involving the Lauricella functions,
generalized polynomials and the multivariable H-function and generalized M-series as given
||Issn 2250-3005 ||
||February||2014||
Page 36

3. Fractional Derivative Associated With The…


    x)



D

M
 
 ,m


y  

N
N M 
1
1



    0 k, M  0
z    y  )}
1


(r)
v
(r)
(r)
D

w      x)}
r
1
r


1    y    1

w      x)}
1

 N 

 M   M
s
S 1
N   N
1
s


1

r    y    r
 N 
M k
1 1
s

k 







w   x)
1

w   x)}
r





   x)
   x)
a
b
1  y 1

a
b
s  y s










M k
s s
k 
1

(r)

z    y   
r
k 0
s
  u' , v' ) ;...; (u
0   3
s


k 0
1
M
s
A  3, C  3 :[B' , D' ] ;...; [B
H


F







   x) 1  y    2  H 





A[N
1
 k  N  k 
1
s
s
s
  
1 y 1 1
1
             
1
1
r
r


s




     a k   k :  

i i
1
1
r 


i 1



 
r y r  1
r
s
s

 


 

 r)
(r)
(r)
    a k   k :  
    k   b k   k :    a) : ',..., 
  b' ) : '; ];...; [(b
 


 
1 1
1
1
r
i i
2
1
r 

 

i 1
i 1

 

s




(r)
(r) (r)
    k   b k   k :                    c) :  ',..., 
  d' ) : ' ] ,..., [d



i i
2
1
r 
1
1
r
r


i 1








…(3.1)
where


  1   1    q  2M) (1  p  q  M)
k ! M ! (1  p  q  M)
 1
k
   k     1  p)
s
  k
1
k
.    x)
. F
M
 z  z 
1
r
k
 
M
 1  p)

  1  1
k
s

a k
i i
i 1
  k  
2
 y)
 c   c 
1
R
(d   d
1
  M)
R

m
R


i 1
b k
i i
t
  
  0  s  0  i  1,2,..., r
i
i
R
and
r
Re(   

i 1
 d (i)
 j

(i)
i 

 j


 1



 d (i) 
 j 
Re (     
 1
(i) 
i 
 
i 1
 j 
Proof. In order to prove (3.1) express the Lauricella function by (2.4) and the multivariable H-function in terms of
Mellin-Barnes type of contour integrals by (2.2) and generalized polynomials given by (2.3) respectively and
r
generalized M-series (2.8) and collecting the power of    x) and (y    Finally making use of the result
(2.7), we get (3.1).
||Issn 2250-3005 ||
||February||2014||
Page 37

4. Fractional Derivative Associated With The…
IV.
PARTICULAR CASES
With  = A = C = 0, the multivariable H-function breaks into product of Fox’s H-function and
consequently there holds the following result


    x)



D

M

 
 ,m

y  
   x)



z    y   )} 1
1


z    y    r
r
y  
1

r
2

H
N
(r)
(r)
3,3 :[B' , D' ] ;...; [B
M


k 0
1
0  3   u' , v' ) ;...; (u
s
s

 N 
1
v
(r)
(r)
D








M k
1 1
k 
k 0
s
u
B
i 1

    0 k, M  0
H


N M 
1
1






F



 M  M
s
S 1
N  N
1
s


(i)
v
(i)
D
(i)
 N 
s

M k
s s
k 
1
w   x)
1

 w i      x)}


(i)


a
b
    x) 1  y    1


    x) a s  y    b s

A[N
1
i
   y   

i
b
(i)

d

(i)

(i)

(i)





 k  N  k 
1
s
s
s
  
1 y 1 1
1
             
1
1
r
r


w   x)}
r





s




     a k   k :  

i i
1
1
r 


i 1


  
r y r r
r
s
s

 


 

(r)
(r)
    a k   k :  
    k   b k   k :     b' ) : '; ];...; [(b



 
1 1
1
1
r
i i
2
1
r 

 

i 1
i 1

 

s




(r) (r)
    k   b k   k :                    d' ) : ' ] ,..., [d



i i
2
1
r 
1
1
r
r


i 1








…(4.1)
valid under the conditions surrounding (3.1).
If 
II.


    x)



D

M


(i)
 ,m

    0 k, M  0
H

 




(i)

 1 (i = 1,2,…) equation (4.1) reduces to
y  
   x)

1


F




z    y   )} 1
1


z    y    r
r
y  

r
2

G
i 1
N
N M 
1
1

k 0
1
0  3   u' , v' ) ;...; (u
3,3 :[B' , D' ] ;...; [B
||Issn 2250-3005 ||
(r)
(r)
s
M


s

 N 
1
k 
k 0
s
v
D
(r)
(r)








w   x)
1
M k
1 1
u
B

 M  M
s
S 1
N  N
1
s


(i)
v
(i)
D
1

w   x)}
r

 w i      x)}


(i)
(i)
 N 


a
b
    x) 1  y    1


    x) a s  y    b s

s
M k
s s
k 
A[N
1





i
   y   

i
b
(i)
d

(i)




 k  N  k 
1
s
s
s
  
1 y 1 1
1
             
1
1
r
r


s




     a k   k :  

i i
1
1
r 


i 1



 
r  y) r  r
r
||February||2014||
Page 38

5. Fractional Derivative Associated With The…
s
s

 


 

(r)
    a k   k :  
    k   b k   k :     b' );...; [(b


1 1
1
1
r  
i i
2
1
r 

 

i 1
i 1

 

s




(r)
    k   b k   k :                    d' ) ,..., [(d


i i
2
1
r 
1
1
r
r


i 1








…(4.2)
valid under the conditions as obtainable from (3.1).
III. Let Ni = 0 (i = 1,…,s), the result in (3.1) reduces to the known result given by Sharma and Singh [ ], after a
little simplification.
IV. Replacing N1,…,Ns by N in (3.1) we have a known result recently obtained by Chaurasia and Singh [ ].
V.
ACKNOWLEDGEMENT
The authors are grateful to Professor H.M. Srivastava, University of Victoria, Canada for his kind help
and valuable suggestions in the preparation of this paper.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
V.B.L. Chaurasia and V.K Singhal, Fractional derivative of the multivariable polynomials, Bull. Malaysian Math. Sc. Soc. (Second
Series), 26 (2003), 1-8.
M. Sharma, Fractional integration and fractional differentiation of the M-series, J. Fract. Calc. and Appl. Anal.,Vol.11, No.2
(2008), 187-191.
M. Sharma and Jain, R., A note on a generalized series as a special function,n of fractional calculus. J. Fract. Calc. and Appl. Anal.,
Vol.12, No. 4 (2009), 449-452.
K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
C.K. Sharma and Singh Indra Jeet, Fractional derivatives of the Lauricella functions and the multivariable H -function, Jñānãbha,
1(1991), 165-170.
H.M. Srivastava, A multilinear generating function for the Konhauser sets of biorthogonal polynomials suggested by the Laguerre
polynomials, Pacific J. Math., 117 (1985), 157-191.
H.M. Srivastava and S.P. Goyal, Fractional derivatives of the H-function of several variables, J.Math. Anal. Appl., 112 (1985),
641-651.
H.M. Srivastava, K.C. Gupta and S.P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian
Publishers, New Delhi-Madras, 1982.
H.M. Srivastava and R. Panda, Some bilateral generating functions for a class of generalized hypergeometric polynomials, J. R eine
Angew. Math. 283/284 (1976), 265-274.
H.M. Srivastava and R. Panda, Certain expansion formulas involving the generalized Lauricella functions, II Comment. Math.Univ.
St. Paul., 24 (1974), 7-14.
H.M. Srivastava and M.C. Daoust, Certain generalized Neuman expansions associated with the Kampé de Fériet function, Nederl.
Akad. Wetensch Indag. Math., 31 (1969), 449-457.
||Issn 2250-3005 ||
||February||2014||
Page 39