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Recently Diophantine findings and conjectures concerning the eigenvalues of certain tridiagonal matrices, and correspondingly the zeros of the polynomials associated with their secular equations, were arrived at via the study of the behavior of certain isochronous manybody problems of Toda type in the neighborhood of their equilibria 1, 2 for a review of these and other analogous results, see 3, Appendix C. To prove some of these conjectures a theoretical framework was then developed 4–6, involving polynomials defined by threeterm recursion relations—hence being, at least for appropriate ranges of the parameters they feature, orthogonal. This result is generally referred to as “Favard theorem,” on the basis of 7; however, as noted by Ismail, a more appropriate name is “spectral theorem for orthogonal polynomials” 8. Specific conditions were identified—to be satisfied by the