Complete Bernstein functions and scaled ultraspherical zeros

Abstract

Let $z_{n,j}(\lambda )$ denote the positive zeros, in decreasing order, of the ultraspherical polynomial $C_n^{\lambda }$, $\lambda >-1/2$, with the reduced limiting interpretation at $\lambda =0$ specified below. Our principal result settles three higher-monotonicity questions of Gautschi: two as printed and the natural open-interval form of the third, whose printed endpoint $\lambda =0$ is singular. For every $n\ge 3$, $$\sqrt{\lambda +1}{z}_{n,j}(\lambda )$$ becomes a complete Bernstein function after translation of its parameter interval to $(0,\infty )$. For every $n\ge 2$, $$\sqrt{\lambda }{z}_{n,j}(\lambda )$$ is a complete Bernstein function on $(0,\infty )$. For every $n\ge 4$, the largest-zero trajectory $$\sqrt{\lambda +\frac{2n^2+1}{4n+2}}{z}_{n,1}(\lambda )$$ has the same property, whilst for every other positive zero the derivative of this scaling fails to be completely monotone. Separately, an exact calculation in degree $4$ provides a counterexample to a fourth conjecture of Gautschi, concerning the linear scaling $\lambda z_{n,j}(\lambda )$.