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
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
Copyright © 2005 Mohammad Masjed-Jamei and Mehdi Dehghan. 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.
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.