Expansion of some well-known functions and classical polynomials in series of the generalized Laguerre polynomials.1 - Ben Hamdin, Haniyah (Libyan Journal of Science &Technology, 2022)
The generalized Laguerre polynomials form a complete set (orthogonal and normalized) in the space ** with respect to a certain weighting function because they are the Eigen functions of a second-order differential operator. Here we shall show how to expand some well-known classical polynomials such as the Legendre and the Hermite polynomials in series of generalized Laguerre polynomials. Since the generalized Laguerre orthogonal polynomials are set of polynomials that are mutually orthogonal to each other with respect to a measure of weighting function that is just the integrand of the gamma function, thus it grants us the guarantee of the ability to expand the first kind of Bessel functions and the gamma function in terms of the generalized Laguerre polynomials. The series expansion in terms of the generalized Laguerre polynomials can be achieved by following various approaches. For instance, the series expansion of the first kind Bessel functions is gained by the generalized hypergeometric function approach, whereas the series expansion of gamma function was obtained directly by the usual way that is by calling the orthogonality and orthonormality properties of the generalized Laguerre polynomials. Another powerful technique to gain the series expansion of the classical polynomials is the generating function approach as it has been followed here to obtain a series expansion of the Legendre and the Hermite polynomials in terms of the generalized Laguerre polynomials. The series expansion in series of the generalized Laguerre polynomials has a variety of applications in mathematics, physics, and engineering