Legendre polynomials are a sequence of orthogonal polynomials that arise in solving problems in physics and engineering, particularly in the context of potential theory and quantum mechanics. They are defined on the interval from -1 to 1 and are denoted as P_n(x) , where n is a non-negative integer. These polynomials can be generated using Rodrigues' formula, which provides a way to express them in terms of derivatives. These polynomials have important properties, such as orthogonality with respect to the weight function 1 on the interval [-1, 1]. This means that the integral of the product of any two different Legendre polynomials over this interval equals zero. They also play a crucial role in spherical harmonics, which are used in various applications, including geophysics and computer graphics.