Laplace Transforms of Classical Orthogonal Polynomials Using the Finite Sum Representations

Classical orthogonal polynomials are Hermite polynomials, Laguerre polynomials, and Jacobi polynomials (also known as hypergeometric polynomials, and the special cases include, Gegenbauer, Legendre, Zernike, and Chebyshev polynomials). Orthogonal polynomials occur across mathematics and many areas of science. Understanding their properties has drawn attention since the beginning of the subject and continues to this day, with many open problems being actively pursued. Orthogonal polynomials can either be studied individually or collectively in a very general setting. In this paper, we shall describe the basic properties which are common to all the orthogonal polynomials, without choosing a specific orthogonal polynomial. This will enable us to get a better understanding of the basic properties possessed by all classical orthogonal polynomials in a very general setting. The basic properties include: orthogonality; expansion of functions in terms of orthogonal polynomials; derivative of orthogonal polynomial sets are again orthogonal polynomial sets; existence of a three-term recurrence relation; existence of generating functions; nature of roots of orthogonal polynomials; the interlacing of roots, finite sum representation and the determinant representation. Then, we shall present the Laplace transforms of each of the aforementioned orthogonal polynomials using different properties in each case. We shall also show that the determinant representation of orthogonal polynomials cannot be used to determine their Laplace transforms. We shall further point out that the finite sum representations of orthogonal polynomials are more convenient to obtain closed-form expressions for their Laplace transforms.