Zeros of ultraspherical polynomials and the Hilbert-Klein formulas (Q5949515)

It is very well known that the Gegenbauer polynomials \(C_n^\lambda(x)\) with \(\lambda>-1/2\), have \(n\)-simple zeros that lies in \((-1,1)\). In the present paper the authors investigate the behaviour of the zeros when \(\lambda\) goes from \(-1/2\) to \(1-n\), being \(n\) the degree of the polynomials. In fact they show how the zeros migrate from \((-1,1)\) to the imaginary axis connecting this with the classical formulas by Hilbert and Klein.