Orthogonality Relations for the Primitives of Legendre Polynomials and Their Applications to Some Spectral Problems for Differential Operators

In this paper, the properties of the primitives of Legendre polynomials on the interval $[0;1]$ are studied. It is proved that the Legendre polynomials form an “almost” orthogonal system. Namely, for a fixed order of the primitive, only finitely many of these polynomials can be nonorthogonal. These properties underly the relationship between the spectral problems for differential operators in $L_2[0;1]$ and the spectral properties of generalized Jacobi matrices.