DETERMINANT REPRESENTATIONS OF LAPLACE TRANSFORMS OF CLASSICAL ORTHOGONAL POLYNOMIALS

Classical orthogonal polynomials are Hermite polynomials, Laguerre polynomials and Jacobi polynomials. They arise in many areas of mathematics and physical sciences. Laplace transform is a powerful mathematical operation that converts complex differential equations to simpler algebraic equations. In this paper, we show that the various properties such as the governing differential equations, and the three-term recurrence relations are not suitable for developing a general approach to the Laplace transforms of classical orthogonal polynomials. We show that the finite sum representation leads to the Laplace transform of classical orthogonal polynomials. We further show that the Laplace transforms thus derived can be expressed using determinants. We explicitly derive the determinant form of the Laplace transforms of five classical orthogonal polynomials.