Integral Equation Involving Bessel Polynomial Suggested by Hermite Polynomial

1. INTRODUCTION

Integral equations and their applications constitute a fascinating area of applied mathematics. The Generalized Hermite polynomial is a powerful tool to solve many integral equations. Many boundary value problems reduced to the problem of solving integral equations whose kernel involves many well known classical polynomials like those of Hermite, Laguerre, Bessel, Legendre, Jacobi etc. During the recent past attempts have been made to generalize these classical polynomials with the help of Rodrigue‘s formulae. To mention Goued Hopper[11] gave a generalization of Hermite polynomials by formulae. and we have used where and ,, and rapare parameters, for suitable value of ,, and rap(1.1) reduced to modified Hermite, modified Laguerre and modified Bessel polynomials. In view of these generalizations it is worth considering integral equations involving as kernel and such we prove the following theorem.

If f is an unknown function satisfying the integral equation. Where and g is a prescribed function then f is given by For

To Prove the Theorem we shall use of Mellin Transform and shell discuss case 0 , 0rp By the convolution theorem for Mellin Transform (2.1) reduces to

When 1 and ,rpb Applying Mellin Transform of equation (3.1) and use the result of Erdelyi [12], we get Where We write equation (3.1) in the form Replacing sby 2sna Where Then from (3.3) and (3.5) By use of definition of H function. We get the inverse transform ()Lxof *()Lsas Where ,,pqmnH are Fox‘s H functions defined by [8]. We get Hence using (3.7) This proves the theorem.

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Ph.D. Scholar, Department of Mathematics, RNT University, Bhopal