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 X 1 -Jacobi and X 1 -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 L 2 Hilbert spaces. Moreover, we prove a converse statement similar to Bochnerʹs theorem for the classical orthogonal polynomial systems: if a self-adjoint second-order operator has a complete set of polynomial eigenfunctions { p i } i = 1 ∞ , then it must be either the X 1 -Jacobi or the X 1 -Laguerre Sturm–Liouville problem. A Rodrigues-type formula can be derived for both of the X 1 polynomial sequences.