Asymptotic Root Dynamics under Repeated and Fractional Integration
Abstract
Fix a polynomial P(x)=∑j=0m ajxj of degree m, and let IkP be its k-fold antiderivative with all integration constants zero. Borwein, Chen, and Dilcher proved that, after x=ky, the nonzero zeros of IkP converge to the nonzero zeros of L[P](y)=∑j aj j! yj. 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 cj(h)=∏ℓ≤ j(1+ℓ h)−1 sampled at h=1/k and h=1/α. We prove rk=kλ+CP(λ)+DP(λ)/k+O(k−2) with explicit CP and DP. Hence λ is at once the scaled zero, the limit of successive root differences, and the continuous-order velocity, each with an explicit correction; CP(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 √k like the zeros of the Hermite polynomial Heq, and for real P the nonzero zeros stay real at every positive order exactly when L[P] is real-rooted.
Related
- Asymptotic behavior of bounded Mild Solutions to some fractional differential equations involving the ψ-Hilfer derivative
- Global front solutions converging to an asymptotic traveling wave in a time-fractional Amari model
- Contour-Based Fast Fractional Integration in a Jacobi Fractional Polynomial Basis
- Asymptotic pseudo-periodicity of solutions for nonlinear fractional evolution equations
- The Kato square root property for fractional elliptic operators in divergence form
BibTeX
@misc{lawless2026asymptotic,
title = {{Asymptotic Root Dynamics under Repeated and Fractional Integration}},
author = {Patrick Lawless},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12498},
eprint = {2610.12498},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12498},
}
RIS
TY - UNPB
TI - Asymptotic Root Dynamics under Repeated and Fractional Integration
AU - Patrick Lawless
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.12498
UR - https://arxiv.org/abs/2610.12498
AB - Fix a polynomial P(x)=∑_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)=∑_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)=∏_ℓ≤ j(1+ℓ 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 √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.
ER -