The research paper published by IJSER journal is about Certain Result involving Special Polynomials and Fractional Calculus 1
ISSN 2229-5518
P.K. Chhajed, D.Nagarajan
2000 Mathematics Subject Classification. Primary 11M06, 11M35; Secondary 33C20.
1. INTRODUCTION AND PRELIMINARIES.
( x, s, a)
1 e at t s 1 1 xet dt
The use of integral transforms to deal with fractional derivatives trace back to Riemann and Liouville
[9,10 ]. The possibility of exploiting integral transforms in a wider
(s) 0
R (a) 0; either x 1, x 1
(1.3)
context involving “ exotic” operator has been discussed in references
[ 4,5 ], where taking advantage form the definition of the generalized
( b ) ( ) function [ 2,p.9 ].
,
and R (s) 0 or x 1R (s) 1
and class of function introduced in [7]
t b
b ( )
t 1e
0
t dt
(1,1)
x, s, a, b
1 at b
t 1e t
1 xet dt
and
R (b) 0; b 0
R 0,
( )
0, 1R (a) 0, R (b) 0; whenR (b) 0
at b
.
ab
( ) a
t 1e
0
t dt
(1.2)
the neither x 1( x 1), R ( ) 0, or x 1, R ( )
It has been shown that
a
b
0; b 0, R (a) 0, R 0 .
d
! f ( x) a
at b
e t
d
t dx f (x)dt
dx ( a,b) 0
Pramod Kumar Chhajed is currently pursing PhD degree program in
Mathematics in Pacific University ,Udaipur, India, Ph.0091-
at b
9351126603,Email: pramod_udaipur@yahoo.com.
Dr. D. Nagarajan is working as Professor in Asan Memorial college of Engineering and Technology, Chennai, India, Email :
a e
0
t f (x ln(t ))dt
(1.5)
dnrmsu2002@yahoo.com
An analogous result can be exploited for more complicated operator,
be recalling, indeed , that
d
The function (x, s, a) [ 1,7 ] has the integral representation
e
We find
dx f (x) f (e x) . (1.6)
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The research paper published by IJSER journal is about Certain Result involving Special Polynomials and Fractional Calculus 2
ISSN 2229-5518
d
at b x d
t b
t
1
t
! f ( x) a e
t t dx f ( x)dt
e t
1 xe
dt
dx ( a,b) 0
at b
(b) ( ) 0
a e
t f ( xt )dt .
( )
x
x, , , b .
0
Furthermore noting form (1.2)
(1.7)
at b
Special case of (1.11)
(b ) ( )
(1.11)
a 1 e
t t 1dt .
(i).
b 0, 1, then
We can also conclude that
( ab) ( ) 0
(1.8)
x 1
( ) x
1 ( x, ,1)
f ( x)
1 b
t 1e t e
t d
dx f ( x)dt
dx 1 x
( )
1
x ( x, ,1) .
dx (b)
(b) 0
1 b
t 1e t f ( x t)dt ,
(1.12)
Where (x, ,1) is the generalized zeta function defined by (1.3) [1,7].
(b) ( ) 0
(1.9)
Case (ii) When
b 0 ,
or
x d
1
1
x (x, , )
x
f ( x)
1 b
t 1e t f (et x)dt .
dx 1 x
dx (b )
(b) ( ) 0
x * (x, , )
(1.10)
,
(1.13)
This last result becomes more interesting by noting that its relevance
which was studied recently by Goyal and Laddha [11].
to the in [7].
x, s, a, b , (x, s, a) and ( ) functions defined
The series representation of
* (x,, ) is
It is indeed readily understood that
xn
* ( x, , ) n
x d
1 1
n 0 n n!
dx (b ) 1 x
(1.14)
b
1, R ( ) 0, 
x 1
.
1 t 1e
t
1
t 1 dt
(b ) ( ) 0
1 e x
After clarifying that the use of meaningless operational form may have non trivial consequence if placed in a proper frame work .We will discuss further integral transform consequence of equation (1.8) , we find that
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The research paper published by IJSER journal is about Certain Result involving Special Polynomials and Fractional Calculus 3
ISSN 2229-5518
f ( x)
2
f ( x)
x (b )
t b t
x2
(b )
b .
1 e t t 1e x f ( x)dt
t
1
x 2
e t
(b ) ( ) 0
t e f ( )d dt
1 4t
2(b ) ( ) 0 t
t b
(1.19)
1
( ) e
(b) 0
or
t t 1 f (x t )dt ,
In this introductory section we have state that by combining the properties of exponential operator and suitable integral representation.
(1.15)
In this paper we will show how families of special polynomial may
provide a powerful complement to the theory of fractional operators.
x
f ( x)
2. INTEGRAL TRANSFORM AND SPECIAL POLYNOMIALS.
x (b )
.
t b
It is well known that the polynomial [ 12 ]
1 e t t 1 f (et x)dt
(b ) ( ) 0
n
yr xn2r
(1.16)
( 2)
n
x, y n! r !(n 2r)! ,
Let us now consider the case involving second order derivatives namely
r 0
(2.1)
2
f ( x)
can be viewed as Gauss Transform of the ordinary monomial
x n , we find indeed,
x2
(b )
.
b 2
( 2) ( , )
1
( x )2
4 y n
t t
e t t 1e x2 f ( x)dt
H n x y
e
d .
(b ) ( ) 0
(1.17)
2 y
(2.2)
The action of the second of the exponential operator containing the
According to equation (1.18) and (2.2) , we obtain
second order derivative can be specified by mean of the Gauss transform
y
2
xn
1 t b
e t t 1H ( 2) ( x, yt )dt
2
t 2
e x f ( x)
1
e
x 2
4t
f ( )d ,
x2
(b )
(b ) 0
n
(2.3)
(1.18)
The transform on the R.H.S of equation (2.3) defines new family
(,2)
so that
of polynomials, denoted by
H(n, ) (x, y, b; ) .
The relevant generating function is easily obtained from their operational definition.
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The research paper published by IJSER journal is about Certain Result involving Special Polynomials and Fractional Calculus 4
ISSN 2229-5518
y
xn
H ( ,2) ( x, y, b; )
H (,2) ( x, y, b; )
x2
( n, )
( n, )
(b )
(2.4)
( 1)
b
H ( ,2) ( x, y, b; )
and the series representation
(2.9)
(b ) ( )
y
2
xn n!
and
2
(b )
n
(b ) ( )
.
y
1
xn
(b ) ( )
H ( ,2) ( x, y, b; )
2 (b)k ( r k ) k r y r xn 2 r
x2
( 1)
( n, )
k 0 r 0
k !r !(n 2r )!
n(n 1)
( 1)
(b )
(2.5)
(b ) y H ( ,2) ( x, y, b; )
( 1) 1 ( n2, )
Special case of (2.5)
(b) ( )
b H (,2) (x, y, b; )
When
b 0 ,
( 1) 1
( n, )
H (,2) (x, y, b; ) H (2)
(x, y; ) ,
(2.10)
the polynomial
P.E. Ricci [4].
H ( 2 ) ( x, y; ) was discuss by G.Dattoli and
Further result will be discussed in the following section.
The properties of this new family of polynomial are fairly easy to obtain and we reported some properties hear.
From definition (2.5) we find .e.g.
3, Extension and Concluding Remarks
We have already remarked that the polynomials
(,2)
( 2)
H (,2) (x, y, b; )
can be recognized as special form of
x H( n, ) ( x, y, b; )
n H n1 ( x, y, b; )
truncated polynomials we will further discuss identifications.
(n, )
H ( ,2) ( x, y, b; )
y
(2.6)
According to the discussion of the previous section, the following identity can easily be proved:
n(n 1)
H ( ,2) ( x, y, b; )
n
1 t b
1 n
n 2
y
x e
t t x yt dt
(2.7)
x (b)
(b) ( ) 0
(,2)
(,2)
b H( n, ) ( x, y, b; )
H n
(2.8)
( x, y, b; )
H (,1) (x, y, b; )
(3.1)
The properties of the polynomials
H (,1) (x, y, b; ) can also
be easily recovered. It is However worth stressing their link with the
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The research paper published by IJSER journal is about Certain Result involving Special Polynomials and Fractional Calculus 5
ISSN 2229-5518
so called Bessel Polynomials [9 ]. By setting
3. Dattoli G., Cesarano C. and sacchetti D., A note on truncated polynomials, Applied Math and Computational
b 0, n 1, x 1, y z
2
, we can make the identification
(appear).
4. Dattoli G., Ricci P.E. and Cesarano C., Fractional derivatives: Integral representation and generalized
polynomials, J. Coner Applic. Math (appear).
5. Dattoli G., Ricci P.E., Cesarano C. and Vaxquez L., Special
(1) z
polynomials and fraction calculus, Math and
n1 H n
1, ,1 yn ( z) ,
(3.2)
Comp.Modelling 37(2003),729-733.
6. Oldham H.and Spainer N., The Fractional Calculus,
Academic Press ,San Diego, CA, 1974.
7. Raina R.K. and Chhajed P.K., Certain results involving a class of functions associated with the hurwitz zeta function. Acta Math.Univ.Comeniance Vol. LXXIII,(2004),89-100.
8. Raina R.K. and Nahar T.S., A note on certain class of
In this concluding section we find it worthwhile to mention briefly
hear a multivariable extension of the class of function
H(n, ) (x, y, b; ) .This multivariable polynomials
(,1) m
function related to hurwitz zeta function and Lambert transform, tamkang J.Math 31(2000),49-55.
9. Rainvile E.D., Special Functions, Macmillian, New York,
1960.
10. Widder D.V., An Introduction to Transform Theory,
H( n, )
x , y, b;
1
can be defined by
Academic Press, New York 1971.
m s 1
xs s
xn
s 2
x1 (b )
(b ) ( )
t b
e t t 1H ( m) ( x x t, x t,..., x t )dt
n 1, 2 3 m
0
( ,1) m
H ( n , )
x , y, b;
1
(3.3)
where
xm
x1 , x2 ,....xm .
Further comment on the properties of the above function and on their usefulness in application will be presented elsewhere.
REFERENCES
1. Andrews L.C., Special Function for Engineering and
Applied Mathematicians, Macmillan ,NewYork,1985.
2. Chaudahry M.Aslam and Zubair Syed M. , On a Class of
Incomplete Gamma Functions with Application, Chapman
& Hall/CRC ( Boca Raton/ London/ New
York/Washington D.C.)2000.
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