Quadratic and cubic transformations and zeros of hypergeometric polynomials (Q1612319)

The authors consider zeros of hypergeometric polynomials \(_2F_1 (-n,b;c;z)\), where \(b\) and \(c\) are real, and the \(_2F_1\) admits a quadratic or a cubic transformation. The quadratic class comprises twelve such polynomials, but there are only four independent theorems because of the linear transformations of \(_2F_1\). Three of these have been given recently by \textit{K. Driver} and \textit{P. Duren} the fourth is treated by the present authors elsewhere [J. Approximation Theory 110, 74-87 (2001; Zbl 0996.33008)]. The cubic class includes sixteen polynomials. Six independent theorems are given; the polynomials involved have \((b,c)\) equal to \((-n\pm \frac 13 ,-2n)\), \((-n\pm \frac 13,-2n \pm \frac 23)\), \((n+1,2n+2)\), and \((\frac 13(-n+1),\frac 13(-2n+2))\), respectively. The remaining cases are obtained as corollaries. Proofs are based upon results in the papers already indicated.