A generalization of formulas for the discriminants of quasi-orthogonal polynomials with applications to hypergeometric polynomials

Abstract

Let $K$ be a field. In this article, we derive a formula for the discriminant of a sequence $\{r_{A,n}+c r_{A,n-1}\}$ of polynomials. Here, $c \in K$ and $\{r_{A,n} \}$ is a sequence of polynomials satisfying a certain recurrence relation that is considered by Ulas or Turaj. There are several works calculating the discriminants of given polynomials. For example, Kaneko--Niiho and Mahlburg--Ono independently proved the formula for the discriminants of certain hypergeometric polynomials that are related to $j$-invariants of supersingular elliptic curves. Sawa--Uchida proved the formula for the discriminants of quasi-Jacobi polynomials. In this article, we present a uniform way to prove a vast generalization of the above formulas. In the proof, we use the formulas for the resultants Res$(r_{A,n},r_{A,n-1})$ by Ulas and Turaj that are generalizations of Schur's classical formula for the resultants.