On certain Gegenbauer polynomial sums (Q2782460)

The author proves two theorems concerning the non-vanishing of Gegenbauer polynomial sums in the unit disk of the complex plane. Theorem 1: Let \(-1<x<1\). For all positive integers \(n\), we have \(\sum^n_{k=0} C^\lambda_k (x)z^k\neq 0\), \(|z|\leq 1\), when \(0<\lambda\leq {1\over 2}\). Theorem 2: Let \(-1<x<1\). For all positive integers \(n\) we have \(\sum^n_{k=0} C_{2k}^\lambda (x)z^k\neq 0\), \(|z|\leq 1\), for \(0< \lambda\leq 1\). These results extend previous ones of Szegő for Legendre polynomials and the reviewer [\textit{S. Ruscheweyh}, SIAM J. Math. Anal. 9, 682-686 (1978; Zbl 0391.30010)]. The author also gives a related system of non-negative Jacobi polynomial sums which can be proved in a similar fashion as the above mentioned theorems (namely using integral presentations which reduce the new cases to known ones).NEWLINENEWLINEFor the entire collection see [Zbl 0980.00021].