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

TL;DR

This paper provides explicit formulas for the resultants and discriminants of classical quasi-orthogonal polynomials, linking algebraic properties to combinatorial and number-theoretic applications, including Diophantine equations and hyperelliptic curves.

Contribution

It generalizes previous results on resultants and discriminants of quasi-orthogonal polynomials and connects these to Diophantine equations and hyperelliptic curve rational points.

Findings

Explicit formulas for resultants and discriminants of quasi-orthogonal polynomials.

Reduction of Diophantine equations to hyperelliptic curve rational point existence.

Nonexistence theorem for solutions of Hausdorff-type equations.

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.