An extended class of orthogonal polynomials defined by a Sturm-Liouville problem

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.