We derive explicit formulas for the discriminants of classical quasi-orthogonal polynomials, as a full generalization of the result of Dilcher and Stolarsky (2005). 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 show a nonexistence theorem for solutions of Hausdorff-type equations by applying our discriminant formula.
The first author was supported in part by Grant-in-Aid for Young Scientists (B) 26870259 and Grant-in-Aid for Scientific Research (B) 15H03636 by the Japan Society for the Promotion of Science (JSPS). The second author was also supported by Grant-in-Aid for Young Scientists (B) 25800023 by JSPS.
"Discriminants of classical quasi-orthogonal polynomials with application to Diophantine equations." J. Math. Soc. Japan 71 (3) 831 - 860, July, 2019. https://doi.org/10.2969/jmsj/79877987