Asymptotic Root Dynamics under Repeated and Fractional Integration

Abstract

Fix a polynomial $P(x)=\sum_{j=0}^m a_jx^j$ of degree $m$, and let $I_kP$ be its $k$-fold antiderivative with all integration constants zero. Borwein, Chen, and Dilcher proved that, after $x=ky$, the nonzero zeros of $I_kP$ converge to the nonzero zeros of $L[P](y)=\sum_j a_j j! y^j$. That convergence determines the leading linear motion but neither the constant offset nor successive differences. We compute the next two terms at a simple nonzero zero $λ$ of $L[P]$. Integer-order and Riemann-Liouville integration reduce, after scaling, to one rational family $c_j(h)=\prod_{\ell\le j}(1+\ell h)^{-1}$ sampled at $h=1/k$ and $h=1/α$. We prove $r_k=kλ+C_P(λ)+D_P(λ)/k+O(k^{-2})$ with explicit $C_P$ and $D_P$. Hence $λ$ is at once the scaled zero, the limit of successive root differences, and the continuous-order velocity, each with an explicit correction; $C_P(λ)$ is the offset of an asymptotic line. For binomials the limiting spacing is a multiple of a gamma-ratio quantity, the Factorial Fraction, and the nonzero reduced zeros form an exact regular polygon at every positive order. For real quadratics the collision order of two branches takes a simple discriminant form. In every degree the collision discriminant carries a universal factor, the one familiar from Laguerre polynomials and supported at negative orders. In the coordinates $(α,x)$ all trajectories lie on one plane curve of degree $m$ with leading form $L[P]$. At a nonzero zero of $L[P]$ of multiplicity $q$, the nearby zeros split on the scale $\sqrt{k}$ like the zeros of the Hermite polynomial $He_q$, and for real $P$ the nonzero zeros stay real at every positive order exactly when $L[P]$ is real-rooted.