Jacobi Endpoint Pencils and Sharp Interlacing for Centered Binomial Samples

Abstract

For an even or odd real entire function $H$ of order at most one, let $B_{2n+1}[H]$ denote its centered binomial sample of odd degree. After removing the zero at $z=\pm 1$ forced by the parity of $H$ and writing $x=z+z^{-1}$, one obtains a real quotient $C_n(x)$. We prove uniform strip conditions on the zeros of $H$ under which $C_n$ and $C_{n+1}$ generate a real-rooted pencil for every $n$. For even $H$ the optimal uniform half-width is $\sqrt{15/28}$, whereas for odd $H$ the half-width $1$ is sufficient. This conclusion is genuinely stronger than separate unit-circle-rootedness of the two sampled polynomials: the latter may hold while the adjacent quotients fail to interlace. The structural result is a theorem for Jacobi spectral multipliers. For every $0<\nu <2$, the quotient problem becomes preservation of the endpoint pencil $y^n(y+t)$. We obtain explicit fixed-$n$ and uniform strip thresholds and determine the exact threshold for $n=1$. The proof combines Bernstein variation diminution with total nonnegativity of finite Jacobi matrices attached to the zero orbits of $H$; a possible unpaired outer real pair, which is not covered by the full defect-class argument, is treated directly on the endpoint pencil. For nonpolynomial even sources satisfying the fixed-$n$ strip condition, a remote-zero-orbit deformation removes common zeros whenever the adjacent images have simple zeros in $(0,4)$. This yields strict interlacing for quotient families associated with Dedekind zeta derivatives and with nested critical-value blocks of self-dual newforms.

62 pages, 2 tables, no figures