LEGENDRE POLYNOMIALS: GENERATING FUNCTIONS AND APPLICATIONS

This study briefly discusses differential equations, types of differential equations, namely, ordinary differential equations and partial differential equations; order and degree of differential equations; methods of solution of first order equations, such as separation of variables, exact, and integrating factors. For second order differential equations, we discuss homogeneous equations and non homogeneous equations. We discuss and mention the characteristics equations and linear combination (linear independence) in the case of second order differential equations with constant coefficients. In the non homogeneous case, we discuss methods of solution, such as, method of undetermined coefficients, method of variation of parameters, and the power series method. This study focuses mainly on second order differential equations with constant and variable coefficients and their methods of solution. These methods include undetermined coefficients, variation of parameters and power series. As a second order ordinary differential equation with variable coefficients, we consider the solution of Legendre’s differential equation using the power series method. Solutions of the Legendre’s equation are known as Legendre polynomials. Generating functions for Legendre polynomials and their applications are explicitly discussed.