Inversion of Integral Equation Associated with Leguerre Polynomial Obtained from Hermite Polynomial


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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; Fox-H 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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  2. Goyal S P and Salim T O., A class of convolution integral equations involving a generalized polynomial set, Proc. Indian

    1

    L(x) H 0,2

    |1n,11 2n,1

    (3.8)

    Acad. Sci. (Math. Sci.): Vol. 108(1), 1998, PP.55-62

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    x

    x

    1 1

    1 1

    2,3 1,11n ,11 2 n,1

    m,n

    m,n

    Where H p,q are Foxs H functions defined by [6].

    general class of polynomials, J. Math, Anal. Appl: Vol 186, 1994, PP.11-20.

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    H m,n x |a1 ,1 ,….,ap , p 1

    (s)xsds

  5. Lala A and Shrivastava P N, Inversion of an integral involving a

    p,q

    b1 ,1 ,….,bq ,q

    2 L

    generalized Hermite polynomial, Indian J. Pure App/. Math. Vol 21, 1990, PP.163-166.

    Where 1, x 0 is a complex variable and

    m n

    m n

    bj j s 1 aj j s

  6. Srivastava H M, Gupta K C and Goyal S P, The H-functions of One and Two Variables with Appilcations, (New Delhi: South Asian Publ.) 1982.

  7. I N Sneden, The use of Integral Transforms, Tata McGraw Hill,

    j 1

    j 1

    New Delhi 1974.

    s

    q p

  8. Srivastava H M and Singhal J P, A class of polynomials defined by

    j m1

    1 bj

    j

    j n1

    aj j s

    generalized Rodrigues formula, Ann. Mat. Pura Appl., Vol 90, 1971, PP.75-85.

  9. Gould H Wand Hopper A T, Operational formulas connected with

    Where m, n, p & q are non negative integers satisfying

    0 n p, 1 m q;

    two generalizations of Hermite polynomials, Duke Math. J. Vol 29, 1962, PP.51-63.

  10. Erdelyi A, Magnus W, Oberhettinger F and Triconi F G, Tables of

j j 1…p and j

j 1…q

Integral Transforms (New York: McGraw-Hili) Vol. I, 1954.

Also we have

Hm,n x |ap , p Hn,m 1 |1bp , p

p,q

bp , p

q, p

x

1ap , p

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