Mathematical Problems in Engineering 
Volume 2005 (2005), Issue 2, Pages 215-230
doi:10.1155/MPE.2005.215

On rational classical orthogonal polynomials and their application for explicit computation of inverse Laplace transforms

Mohammad Masjed-Jamei1,2 and Mehdi Dehghan3

 1Center of Research and Studies, Sanjesh Organization, Ministry of Science and Technology, Tehran, Iran
2Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran
3Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, P.O. Box 15875-4413, Tehran, Iran
Received 26 June 2004; Revised 21 August 2004

Abstract

From the main equation (ax2+bx+c)y″n(x)+(dx+e)y′n(x)−n((n−1)a+d)yn(x)=0, n∈ℤ+, six finite and infinite classes of orthogonal polynomials can be extracted. In this work, first we have a survey on these classes, particularly on finite classes, and their corresponding rational orthogonal polynomials, which are generated by Mobius transform x=pz−1+q, p≠0, q∈ℝ. Some new integral relations are also given in this section for the Jacobi, Laguerre, and Bessel orthogonal polynomials. Then we show that the rational orthogonal polynomials can be a very suitable tool to compute the inverse Laplace transform directly, with no additional calculation for finding their roots. In this way, by applying infinite and finite rational classical orthogonal polynomials, we give three basic expansions of six ones as a sample for computation of inverse Laplace transform.