Zeros of the Hypergeometric Polynomials F(−n, b; −2n; z)

AbstractWe investigate the location of the zeros of the hypergeometric polynomial F(−n, b; −2n; z) for b real. The Hilbert–Klein formulas are used to specify the number of real zeros in the intervals (−∞, 0), (0, 1), or (1, ∞). For b>0 we obtain the equation of the Cassini curve which the zeros of wnF(−n, b; −2n; 1/w) approach as n→∞ and thereby prove a special case of a conjecture made by Martı́nez-Finkelshtein, Martı́nez-González, and Orive. We also present some numerical evidence linking the zeros of F with more general Cassini curves