Interlacing of zeros of Gegenbauer polynomials of non-consecutive degree from different sequences (Q663265)

The author proves that Stieltjes interlacing occurs for the zeros of the Gegenbauer (or ultraspherical) polynomials \(C_{n+1}^\lambda\) and \(C_{n-1}^{\lambda +t}\), where \(\lambda > \frac12\), provided \(t\in (0,k+1]\), and for the zeros of \(C_{n+1}^\lambda\) and \(C_{n-2}^{\lambda +t}\), provided \(t\in \{0,1,2,3\}\). Furthermore, the Stieljes interlacing property for zeros is also proven for \(C^\lambda_{n+1}\) and the \(k\)-th derivative of \(C_n^\lambda\), \(k\in \{1, 2, \ldots, n-1\}\).