文摘
We present two infinite sequences of polynomial eigenfunctions of a Sturm–Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as X1-Jacobi and X1-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval [−1,1] or the half-line [0,∞), respectively, and they are a basis of the corresponding L2 Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second-order operator has a complete set of polynomial eigenfunctions , then it must be either the X1-Jacobi or the X1-Laguerre Sturm–Liouville problem. A Rodrigues-type formula can be derived for both of the X1 polynomial sequences.
