Monomial Moving Zeros for the Heat Semigroup: An Explicit Finite-Order Differential Hierarchy

Abstract

Let $P_t=\exp(t\D^2/2)$ be the heat semigroup. For every integer $k\ge 2$ and parameter $ρ\in\C$, we construct an entire function ${\Gamma }_{\rho ,k}$ whose heat evolution has the prescribed moving zero $x=\rho t^k$ and constant spatial flux there: \[ (P_tΓ_{ρ,k})(ρt^k)=0, \qquad \partial_x(P_tΓ_{ρ,k})(ρt^k)=1. \] The initial function is explicit, \[ Γ_{ρ,k}(x) =x\sum_{q\geq0}\frac{(-2^kρx^{2k-1})^q}{q!} \frac{(kq)!}{((2k-1)q+1)!}, \] and is a generalized hypergeometric function whose entire-function order is strictly below two. Most importantly, it is annihilated by an explicit polynomial-coefficient differential operator of order $2k-1$. Thus every monomial moving boundary admits a finite-order stationary realization after a natural boundary-flux selection, even though the naive order-$k$ family $\D^k+a x^{k-1}$ already fails to produce the exact boundary at $k=4$. No minimality claim is made: the result gives a universal finite-order hierarchy, while the least possible differential order remains open and depends on the choice made inside the inverse kernel.