On the families of orthogonal polynomials associated to the Razavy potential

We show that there are two different families of (weakly) orthogonal polyno mials associated to the quasi-exactly solvable Razavy potential V(x) = (zet a cosh2x - M)(2) (zeta > 0, M is an element of N). One of these families en compasses the four sets of orthogonal polynomials recently found by Khare a nd Mandal, while the other one is new. These results are extended to the re lated periodic potential U(x) = -(zeta cos 2x - M)(2), for which we also co nstruct two different families of weakly orthogonal polynomials. We prove t hat either of these two families yields the ground state (when M is odd) an d the lowest-lying gaps in the energy spectrum of the latter periodic poten tial up to and including the (M - 1)th gap and having the same parity as M - 1. Moreover, we show that the algebraic eigenfunctions obtained in this w ay are the well known finite solutions of the Whittaker-Hill (or Hill's thr ee-term) periodic differential equation. Thus, the foregoing results provid e a Lie-algebraic justification of the fact that the Whittaker-Hill equatio n (unlike, for instance, Mathieu's equation) admits finite solutions.