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

From the main equation ( a x 2 +bx+c ) y ″ n ( x )+( dx+e ) y ′ n ( x ) n( ( n 1 )a+d ) y n ( x )=0 , n∈ + , 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=p z 1 +q , p≠0 , q∈ . 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.