Jacobi Polynomial Expansions

The differential operator generated by the Jacobi differential equation (1 − x2) y″ + [β − α − (α + β + 2)x]y′ + n(α + β + n + l) y = 0, x ∈ [− 1, 1]is considered for all α and β in both the right and left definite spaces. Shifted Jacobi operators when α < 1, β > − 1, when α > − 1, β < 1, and when α < 1, β <1, and the classical Jacobi operator with α > − 1, β > − 1 are introduced. We show that all Jacobi operators are self-adjoint in both spaces. The spectral resolutions of shifted Jacobi differential operators are given by comparing them to the classical Jacobi polynomial expansion.