Jacobi Endpoint Pencils and Sharp Interlacing for Centered Binomial Samples

Abstract

We study interlacing between real polynomial quotients obtained from centered binomial samples of a fixed even or odd real entire function at consecutive odd degrees. For sources of order at most one, we determine the optimal vertical-strip half-widths guaranteeing weak interlacing for every adjacent pair: $\sqrt{15/28}$ for even sources and the explicit algebraic constant $h_{\ast }\approx 1.0656$ for odd sources. These constants are forced by the first nontrivial pair and suffice at every degree. Under the same sharp bounds, nonpolynomial sources have simple zeros in $(-2,2)$ and strict interlacing at every positive index. We also prove an eventual result: if a nonpolynomial source has growth order less than two and only finitely many zeros off the imaginary axis, then all sufficiently high adjacent quotients have simple interior zeros and strictly interlace; the starting index depends only on the finite factor formed from those zeros. The proof turns the sampling problem into an interval real-rootedness problem for a Jacobi differential operator. For fixed degree, we also determine the maximal range of scalar shifts preserving real-rooted polynomials with at most one zero outside the interval. Applications give strict interlacing for quotients attached to completed Dedekind zeta derivatives and to successive centered blocks of critical values of self-dual newforms satisfying explicit spacing and parity conditions.

arXiv comment64 pages. Substantially revised and expanded: new and strengthened sharp, strict, and eventual interlacing results; a broader Jacobi endpoint-preservation theorem; arithmetic applications; and improved proofs and exposition