Zeros of ultraspherical polynomials and the Hilbert–Klein formulas

The orthogonality of the ultraspherical polynomials Cnλ(z) for λ>−12 ensures that all of their zeros are in the interval (−1,1). In a previous paper (Driver and Duren, Indag. Math. 11 (2000) 43–51), we have shown that when λ<1−n, all of the zeros lie on the imaginary axis. Our purpose is now to describe the trajectories of the zeros of Cnλ(z) as λ decreases from −12 to 1−n. In particular, the pattern of migration from the interval (−1,1) to the imaginary axis serves to confirm and “explain” the classical formulas of Hilbert and Klein for the number of zeros of Cnλ(z) lying in each of the real intervals (−∞,−1), (−1,1), and (1,∞).