 Open Access
 Total Downloads : 37
 Authors : Priyank Jain , Dr. Archana Lala , Dr. Chitra Singh
 Paper ID : IJERTV8IS040152
 Volume & Issue : Volume 08, Issue 04 (April – 2019)
 Published (First Online): 09042019
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Inversion of Integral Equation Associated with Leguerre Polynomial Obtained from Hermite Polynomial
Priyank Jain1
Ph.D. Scholar, Dept. of Mathematics Ravindranath Tagore University Bhopal, India
Dr. Chitra Singp
Department of Mathematics
Dr. Archana Lala2 Department of Mathematics S R Group of Institutions Jhansi, India
Ravindranath Tagore University Bhopal, India
AbstractThe objective of this paper is to obtain a solution of certain integral equation whose kernel involves modified Leguerre Polynomial obtained from Generalized Hermite Polynomial by choosing suitable parameters. We believe that many more polynomial like Jacobi, Bessel etc. can be obtained
II. THEOREM
If f is an unknown function satisfying the integral equation.
x dy
by considering suitable parameters involved in Generalized
g x k f ( y) , x 0
(2.1)
Hermite Polynomial. For the sake of illustration we choose suitable parameters involved in Generalized Hermite
0 y y
x 1
Polynomial in place of modified Legurre Polynomial.
Where k(x) x .e .Hn x,,1
(2.2)
KeywordsGeneralized Hermite Polynomial; Mellin Transform; Convolution Theorem; FoxH function.
and g is a prescribed function then f is given by For
r 1, p 1
I. INTRODUCTION
f (x) 1n xn
y
y
H 0,2 1n,11 2n,1
0 2,3
2 1
Integral Equations appears in the most applied areas and are as important as differential equations. Many boundary value problems can be converted into the problem of solving integral equations whose kernel involves many well known classical polynomials like those of modified Hermite, modified Legurre, modified Jacobi etc. Many
x
d n
dy
dy
1,11n ,11 1 n,1
.
.
y g y dy
y
attempts have been made in the past to generalize and unify these classical polynomials with the help of Rodrigues formulae. To mention Gould Hopper [9] gave a
III. SOLUTION
To Prove the Theorem we shall use of Mellin Transform defined by Sneddon [7]
generalization of Hermite polynomials by formula.
H r x, a, p (1)n xaepxr Dn xae pxr
(1.1)
* s1
* s1
f (s) f x .x dx M f (x), s
0
(3.1)
n
and the convolution theorem for Mellin Transform
and we have used
x dy
H 1 x, ,1 (1)n x ex Dn x e x
(1.2)
M k f ( y) , s k* s f * (s)
(3.2)
n
0 y y
where
D d and
dx
r, a, and p are parameters, for
Thus equation (2.1) reduces to
k*(s) f *(s) g*(s)
(3.3)
suitable value of
r, a, and p (1.1) reduced to modified
Where
k* s, f * s, g* s
are respective Mellin
n
n
Hermite, modified Laguerre and modified Bessel polynomials. In view of these generalizations it is worth considering integral equations involving H1 x,,1 as
Transform of k x, f x, g x.
When r 0 and p 0, and in particular
kernel and such we prove the following theorem.
r 1 and p 1,
Applying Mellin Transform to equation (2.2) we get
k*(s) 1n M Dn x .e x ; s ,
and use the result of Erdelyi [10], we get
s 
s n 

s n 
s 
s n 

s n 
k* x
Where
Re(s) n,
where
Re( ) 0
(3.4)
Now taking inverse Mellin Transform on both sides of (3.5), using convolution theorem and result of Mellin Transform
We get
M 1 f * (s n )
s
Re(s) n Re( ),
where
Re( ) 0
M 1 1n L* (s) (1)n g* s n
We write equation (3.3) in the form
g* (s)
f * (s)
s n
n
k* (s)
x n f (x) 1n
x d
y dy
y dy
L
y g y dy
replacing s by s n
f * s n
0
y
x d n dy
s
f (x) 1n xn
L
y g y
1n L* (s) 1n .g* s n
(3.5)
0 y dy
y
s n
Where
Hence using (3.8)
s n
n n
y
y
0,2
1n,11 2n,1
L* (s)
(3.6)
f (x) 1 x
H2,3 
2 1
s .k*
s n
0 x
1,11n ,11 1 n,1
Then from (3.4) and (3.6)
d n
dy
s 2n 
s n 

s 
s n 
s 2n 2 
s 2n 
s n 

s 
s n 
s 2n 2 
L* (s)
(3.7)
dy
dy
This proves the theorem.
y g y .
y
Re(s) n,
where
Re( ) 0
REFERENCES
Re(s) n Re( ), where Re( ) 0
By use of definition of H function. We get the inverse transform L(x) of L* (s) as

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(s)xsds

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m n
m n
bj j s 1 aj j s

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j 1
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s
q p

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j
j n1
aj j s
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j j 1…p and j
j 1…q
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Hm,n x ap , p Hn,m 1 1bp , p
p,q
bp , p
q, p
x
1ap , p