A monotonic property for the zeros of ultraspherical polynomials

Abstract:

It is shown that $\lambda x_{n,k}^{(\lambda )}$ increases as $\lambda$ increases for $0 < \lambda < 1,k = 1,2, \ldots ,\left [ {\tfrac {n}{2}} \right ]$, where $x_{n,k}^{(\lambda )}$ is the $k{\text {th}}$ positive zero of ultraspherical polynomial $P_n^{(\lambda )}(x)$.