Integral of Legendre polynomials and its properties

This paper is concerned with deriving a new system of orthogonal polynomials whose inflection points coincide with their interior roots, primitives of Legendre polynomials. We show a connection between these orthogonal polynomials and two special cases of Jacobi polynomials, we demonstratecertain identities and extremal properties involving both the integral Legendre polynomials. We use mathematical induction to establish the relation between them. We also present some results for these orthogonal polynomials by using some properties of Jacobi polynomials. General expressions are found for the kernels polynomials associated to integral Legendre polynomials, particularly by using the Jacobi polynomials we prove some results and comparisons and applications with kernel integral Legendrepolynomials. These kernel polynomials can be used to describe the approximation of continuous functions by integral Legendre polynomials. The results are then applied to find the minimum value and the minimizing function for various definite integrals. We conclude the paper with some results by using polynomials integrals of the kernels polynomials.

کلیدواژه ها:

Legendre polynomials ، Kernels polynomials ، Christoffel-Darboux formula ، Three-term recursive relation