Fredholm Type Integral Equations with Aleph-Function and General Polynomials Rinku

This paper is devoted to the useful method of solving the one-dimensional integral equation of Fredholm type. The Mellin transform technique for solving a general fredholm type integral equation with the function and a generalized polynomial in the kernel is considered. By specializing the coefficients and various parameters in the generalized polynomials and function, our main theorem would readily yield several results involving simpler kernels.


Introduction
In the last several years a large number of Fredholm type integral equationsinvolving various polynomials or special functions as their kernels have beenstudied by many authors notably Buchman [10], Higgins [11], Love ([3] and [2), Prabhakar and Kashyap [12], Srivastava and Raina [6] and others.In the present paper , we obtain the following one dimensional Fredholm integral equation (1.1 ) involving the Յ -function and a generalised polynomials in the kernel can be solved by resorting to the application of Mellin transforms.
R extends from γ -i∞ to γ +i∞, and is such that the poles, assumed to be simple, of The parameters Aj, B j , A ji , B ji > 0 and aj, bj a ji , b ji ∈ C. The empty product in (1.3) is interpreted as unity.The existence conditions for the defining integral (1.1) are given below: (1.7) The general polynomials of R variables given by Srivastava [4] defined and represented as : Where m i is an arbitrary positive integer and coefficients are arbitrary constants, real or complex.
Let ‫‬ denote the space of all functions f which are defined on R + = [0 , ∞) and satisfy For correspondence to the space of good functions defined on the whole real line (-∞ , ∞) see (Lighthill) [8].The Riemann -Liouville fractional integral (of order µ) is defined by (1.10)

PRELIMINARY RESULTS
We first prove the following result which will be required in proving theorem 1 below.
Let [ ] To prove Lemma 1, we first use the definition of Weyl fractional integral given in (1.10) express the Յ-function and a generalized polynomial, then we change the order of summations and integration (which is justified under the stated conditions), evaluate the t-integral and reinterpreting the resulting Mellin-Barnes contour integral in terms of the Յ -function, we easily arrive at the desired result.
Theorem 1 -Under the sufficient conditions (i), (ii), (iii) and (iv) of Lemma [ ] Proof: Let η denote the first member of the assertion (2.2).Then using Lemma 1 and applying (1.10), we have Assuming the inversion of the order of integration to be permissible just as in the proof of Lemma 1. Now, by definition (1.9), (2.4) gives Which is the second member of (2.2).

SOLUTION OF A FREDHOLM TYPE INTEGRAL EQUATION
To obtain the solution of a fredholm type integral equation (1.1), we use the Mellin Transform technique. ) provided further that max{ Re [(a l -1)/Φ]} < Re [(p(s 1 +…+s R )+k 1 )/q] < min { Re (b j /B j ), (j=1,…m) and (l = 1,… n) Proof: On replacing f by D α-β {f} in (3.1) and applying (2.1), we have Multiplying both the sided of (3.3) by x h -1 and integrating with respect to x from 0 to ∞, we have [ ] where we have assumed the absolute ( and uniform) convergence of the integrals involved, with a view to justifying the inversion of the order of integration.Now evaluate the inner integral in (3.4) by a simple change of variables in the familiar results (c.f., for example, [ 5] and [ 7]), eq.(3.4) reduces to ( )

−
As the solution of the integral equation (3.1) [ 9]Aleph Յ-function occurring in (1.1) introduced by Sudland et al.[ 9]which is defined as a contour integral of Mellin Barnes Type: