Discriminants of classical quasi-orthogonal polynomials, with combinatorial and number-theoretic applications

Abstract

We derive explicit formulas for the resultants and discriminants of classical quasi-orthogonal polynomials, as a full generalization of the results of Dilcher and Stolarsky (2005) and Gishe and Ismail (2008). We consider a certain system of Diophantine equations, originally designed by Hausdorff (1909) as a simplification of Hilbert's solution (1909) of Waring's problem, and then create the relationship to quadrature formulas and quasi-Hermite polynomials. We reduce these equations to the existence problem of rational points on a hyperelliptic curve associated with discriminants of quasi-Hermite polynomials, and thereby show a nonexistence theorem for solutions of Hausdorff-type equations.