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

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.