Asymptotic Root Dynamics under Repeated and Fractional Integration

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 zeros of $L[P](y)=\sum_j a_j j! y^j$. That fixes a directio