On the behavior of Gegenbauer polynomials in the complex plane (Q1759053)

A real entire function \(f\) is said to be in the the Laguerre-Pólya class, denoted by \(\mathcal L\)-\(\mathcal P\), if \(f\) is the uniform limit, on compact subsets of \(\mathbb C\), of polynomials all of whose zeros are real. If \(f\in \mathcal L\text{-}\mathcal P\), then it is known that \[ |f(x+iy)|^2=\sum_{k=0}^{\infty} L_k(f; x)y^{2k},\quad x, y \in \mathbb R, \] where the coefficients, \(\{L_k\}\) are representable as non-linear differential operators acting on \(f\). By a classical result of Jensen, \(L_k(f; x)\geq 0\) for \(f\in \mathcal L\text{-}\mathcal P\) and for all \(x\in \mathbb R\). After proving several interesting preparatory results pertaining to the properties of \(L_k(f; x)\) (Section 2), the authors establish the following main result. Let \(f=P_n^{\lambda}\), \(\lambda >-1/2\), denote the \(n^{th}\) Gegenbauer polynomial, where \(n\geq 2\). Then \(L_k(f; x)\) is strictly decreasing on \((-\infty,0]\) and strictly increasing on \([0, \infty)\) for \(k=1,\dots, n-1\) (Theorem 3.1).