On the Behavior of Gegenbauer Polynomials in the Complex Plane

Abstract

It is well-known that the squared modulus of every function f from the Laguerre–Polya class $${\mathcal{L}-\mathcal{P}}$$ of entire functions obeys a MacLaurin series representation $$|f(x+i y)|^2=\sum_{k=0}^{\infty} L_k(f;x)\,y^{2k}, \quad x,y\in\mathbb{R}$$ , which reduces to a finite sum when f is a polynomial having only real zeros. The coefficients {L k } are representable as non-linear differential operators acting on f, and by a classical result of Jensen L k (f;x) ≥ 0 for $${f\in \mathcal{L}-\mathcal{P}}$$ and $${x\in \mathbb{R}}$$ . Here, we prove a conjecture formulated by the first-named author in 2005, which states that for $${f=P_n^{(\lambda)} }$$ , the n-th Gegenbauer polynomial, the functions $${\{L_k(f;x)\}_{k=1}^{n}}$$ are monotone decreasing on the negative semi-axis and monotone increasing on the positive semi-axis. This result pertains to certain polynomial inequalities in the spirit of the celebrated refinement of Markov’s inequality, found by R. J. Duffin and A. C. Schaeffer in 1941.