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

In this article, we extend the classical framework for computing discriminants of special quasi-orthogonal polynomials from Schur's resultant formula, and establish a framework for computing discriminants of a sufficiently broader class of polynomials from the resultant formulas that are proven by Ulas and Turaj. More precisely, we derive a formula for the discriminant of a sequence { r A , n + c r A , n - 1 } \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{r_{A,n}+c r_{A,n-1}\}$$\end{document} of polynomials. Here, c is an element of a field K and { r A , n } \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{r_{A,n}\}$$\end{document} is a sequence of polynomials satisfying a certain recurrence relation. There are several works computing 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 and applied it to prove the nonexistence of certain rational quadrature formulas. Our main theorem presents a uniform way to prove a vast generalization of the above formulas for the discriminants.