Orthogonality of polar Legendre polynomials and approximation

Let $\{Q_{n}(x)\}$ be a system of integral Legendre polynomials of degree exactly n,and let $\{P_{n}(x)\}$ be polar polynomials primitives of integral Legendre polynomials. We derive some identities and relations and extremal problems and minimization involving of polar integral Legendre polynomials.