On zero distribution of the sum of Gegenbauer polynomials

Abstract

We study the zero distribution of the sum of the first $n$ Gegenbauer polynomials \[ S_n^{(α)}(z)=\sum_{k=0}^n C_{k}^{(α)}(z). \] We prove that for $\alpha >1$, the zeros of $S_n^{(\alpha )}(z)$ lie on the interval $(-1,1)$ for all large $n$.