﻿Solution of Some Integral Equations Involving Confluent <i>k</i>-Hypergeometric Functions

Applied Mathematics
Vol.4 No.7A(2013), Article ID:33977,3 pages DOI:10.4236/am.2013.47A003

Solution of Some Integral Equations Involving Confluent k-Hypergeometric Functions

Shahid Mubeen

Department of Mathematics, University of Sargodha, Sargodha, Pakistan

Email: smjhanda@gmail.com

Copyright © 2013 Shahid Mubeen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received April 19, 2013; revised May 20, 2013; accepted May 28, 2013

Keywords: Linear Integral Equations; Fractional Integrals; Confluent Hypergeometric Functions

ABSTRACT

The principle aim of this research article is to investigate the properties of k-fractional integration introduced and defined by Mubeen and Habibullah , and secondly to solve the integral equation of the form , for , where is the confluent k-hypergeometric functions, by using k-fractional integration.

1. Introduction

Erdélyi  investigated the solutions of integral equations whose kernels contain Legendre functions. Love  solved the integral equations involving hypergeometric functions using fractional derivatives. Using variance of fractional integration, Habibullah  investigated the solution of the integral equations involving confluent hypergeometric functions and Srivastava  discussed the equations with polynomial kernels.

Diaz et al. [6-8] have introduced k-gamma and k-beta functions and proved a number of their properties that we are interested in. They have also studied k-zeta function and k-hypergeometric function based on Pochhammer k-symbols for factorial functions. These studies were then followed by works of Mansour , Kokologiannaki , Krasniqi [11,12] and Merovci  elaborating and strengthening the scope of k-gamma and k-beta functions. Very recently, Mubeen and Habibullah  gave a simple and useful integral representations of generalized khypergeometric and confluent k-hypergeometric functions that could helpful in completing the present research paper.

2. Fractional Integration

Mubeen and Habibullah  defined a k-fractional integration as a variant of Riemann-Liouville fractional integral as for . It reduces to the classical Riemann-Liouville fractional integral by taking as .

3. k-Hypergeometric and Confluent

k-Hypergeometric Differential Equations

The following k-hypergeometric function defined by Mubeen and Habibullah is the solution of the linear second order differential equation of the form .

In this article, we call it k-hypergeometric differential equation. It reduces to ordinary hypergeometric differential equation by taking .

And also the following confluent k-hypergeometric function defined by Mubeen and Habibullah is the solution of the linear second order differential equation of the form .

In this article, we call it confluent k-hypergeometric differential equation. It reduces to ordinary hypergeometric differential equation by taking .

4. Main Results

Theorem 4.1. If  Proof. Consider Put in the above equation, then we get the desired result.

Theorem 4.2. Let for If is a given function, then .

Proof. Set where Apply on both sides, we get the following Changing the order of integration by using Fubini’s theorem. By Theorem 4.1, we have This implies that Since      This may be written as Since , we obtain     This is the solution of the integral equation, if it exists.

This integral equation implies that Now, we find a solution of another integral equation for Theorem 4.3. Let for If is a given function, then Proof. Consider Using the Mubeen’s relation we obtain the following Thus, if then Also, we have the following result 5. Acknowledgements

The author would like to express profound gratitude to referees for deeper review of this paper and their valuable advice and the referee’s useful suggestions that led to an improved presentation of the paper. The author is also pleased to pay special thanks to Dr. Atiq ur Rehman for his support in this research.

REFERENCES

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