Trajectories of the zeros of hypergeometric polynomials \(F(-n, b; 2b; z)\) for \(b< -\frac{1}{2}\) (Q5937036)

In a previous paper [Indag. Math., New Ser. 11, No.~1, 43-51 (2000; Zbl 0969.33003)] the authors have shown that all the zeros of \[ F(-n, b; 2b; z)=1+\sum_{k=1}^n (-1)^k \binom{n}{k} \frac{(b)_k}{(2b)_k} z^k , \] are simple and lie on the circle \(|z-1|=1\) for \(b>-1/2\). The key to this fact was a connection of \(F(-n, b; 2b; z)\) with Gegenbauer polynomials. Using an alternative formula and standard quadratic transformation for hypergeometric functions, it is shown now that for \(b < 1-n\) all zeros are in \((1,+\infty)\); a quantitative result is also provided. The core of the paper is a thorough description (graphically illustrated) of the trajectories followed by the zeros as \(b\) descends from \(-1/2\) to \(1-n\). The ``pathological'' cases of multiple zeros (at \(z=0\) or \(z=1\)) for certain values of \(b\) are also discussed.