Real zeros of 2F1 hypergeometric polynomials

TL;DR

This paper characterizes the parameter values for which certain hypergeometric polynomials have all real, simple zeros and identifies the intervals where these zeros are located, using division algorithms and properties of Jacobi polynomials.

Contribution

It introduces a method based on the division algorithm to find all parameter values leading to real, simple zeros of hypergeometric polynomials and applies quasi-orthogonality of Jacobi polynomials to locate these zeros.

Findings

Identifies parameter conditions for real, simple zeros of hypergeometric polynomials.

Determines intervals on the real line where zeros are located.

Uses division algorithm and quasi-orthogonality techniques.

Abstract

We use a method based on the division algorithm to determine all the values of the real parameters $b$ and $c$ for which the hypergeometric polynomials $_2F_1(-n, b; c; z)$ have $n$ real, simple zeros. Furthermore, we use the quasi-orthogonality of Jacobi polynomials to determine the intervals on the real line where the zeros are located.