Prove entireness of the linear-combination eigenfunctions for all N

Establish rigorously that the special linear combinations of solutions defined in equations (eq:eigenfeven) for even N and (eq:eigenfodd) for odd N exhibit complete pole cancellation and are entire functions of x for generic values of the parameters h_k and the energy, for arbitrary N.

Background

Each individual saddle solution of the finite-difference equation has poles at x = a_I + \hbar n. The authors construct explicit linear combinations (eqs. (eq:eigenfeven) and (eq:eigenfodd)) that are designed to cancel these poles and yield entire off-shell eigenfunctions for generic parameters, which become L^2-normalizable only on a discrete energy spectrum.

They report extensive checks for N=2–6 up to several orders in the \Lambda expansion but note the lack of a rigorous proof that entireness holds generally. A formal proof would validate the exact solvability claim for all polynomial potentials considered.