Asymptotic distributions of the zeros of a family of hypergeometric polynomials (Q2845049)

The authors consider the distribution of zeros for hypergeometric polynomials NEWLINE\[NEWLINE _2F_1(-n,a;-n+b;z), NEWLINE\]NEWLINE where \(n\) denotes a nonnegative integer and \(a,b\) fixed noninteger complex numbers. The main result says that for \(n \to \infty\), the zeros of these polynomials approach the unit circle. The proofs are based on the observation that by careful elementary estimates of the coefficients, these polynomials can be proved to converge inside the unit disc to \((1-z)^{-a}\). Classical transformation formulae like those of Kummer type allow to apply the result also to other sequences of special functions like e.g. Jacobi polynomials. The results are illustrated by twelve pictures of zero sets of special functions.