Elements of Pólya-Schur theory in the finite difference setting (Q2821743)

Let \(\mathcal{HP} \subset \mathbb{R}[x] \) be the set of all real-rooted polynomials (including constant polynomials). The elements of \(\mathcal{HP}\) are also referred to as hyperbolic polynomials. Denote by \(\mathcal{HP}_{\geq\alpha}\) the set of all real-rooted polynomials whose mesh (the minimal distance between the roots) is at least \(\alpha>0\).NEWLINENEWLINEA linear finite difference operator \(T:\mathbb{C}[x]\to \mathbb{C}[x]\) of the form NEWLINE\[NEWLINE T(p)(x)=q_0(x)p(x)+q_1(x)p(x-1) +\dots+ q_k(x)p(x-k) NEWLINE\]NEWLINE where \(q_0(x),\dots,q_k(x)\) are fixed complex- or real-valued polynomials, is called a discrete hyperbolicity preserver if it preserves \(\mathcal{HP}_{\geq 1}\).NEWLINENEWLINE\noindent The authors attempt to develop an analog of \textit{G. Pólya} and \textit{I. Schur}'s theory [J. Reine Angew. Math. 144, 89--113 (1914; JFM 45.0176.01)] in the setting of linear finite difference operators.NEWLINENEWLINE\noindent They prove the finite difference version of the classical Hermite-Poulain theorem.NEWLINENEWLINEA linear finite difference operator \(T(p)(x)=a_0 p(x) + a_1 p(x-1)+ \dots + a_kp(x-k)\) with constant coefficients \(a_0,\dots,a_k\) is a discrete hyperbolicity preserver if and only if all zeros of its symbol polynomial \(Q(t)=a_0 + a_1 t + \dots + a_k t^k\) are real and non-negative.NEWLINENEWLINEThey also present several results about discrete multiplier sequences and operators.