Complete Bernstein functions and scaled ultraspherical zeros

Abstract

Let $z_{n,j}(λ)$ denote the positive zeros, in decreasing order, of the ultraspherical polynomial $C_n^λ$, $λ>-1/2$, with the reduced limiting interpretation at $λ=0$ 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 $λ=0$ is singular. For every $n\geq3$, $$ \sqrt{λ+1}\,z_{n,j}(λ) $$ becomes a complete Bernstein function after translation of its parameter interval to $(0,\infty)$. For every $n\geq2$, $$ \sqrtλ\,z_{n,j}(λ) $$ is a complete Bernstein function on $(0,\infty)$. For every $n\geq4$, the largest-zero trajectory $$ \sqrt{λ+\frac{2n^2+1}{4n+2}}\,z_{n,1}(λ) $$ 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 $λz_{n,j}(λ)$.