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A second order finite-difference equation has two linearly independentsolutions. It is shown here that, like in the continuous case, at most one ofthe two can be a polynomial solution. The uniqueness in the classicalcontinuous Heine-Stieltjes theory is shown to hold under broader hypothesesthan usually presented. A difference between regularity condition anduniqueness is emphasized. Consistency of our uniqueness results is also checkedagainst one of the Shapiro problems. An intrinsic relation between theHeine-Stieltjes problem and the discrete Bethe Ansatz equations allows one toimmediately extend the uniqueness result from the former to the latter. Theresults have implications for nondegeneracy of polynomial solutions of physicalmodels.

On uniqueness of Heine-Stieltjes polynomials for second order finite-difference equations