Rodrigues' descendants of polynomials and Boutroux curves

Consider an arbitrary univariate polynomial $P(x)$ of degree $d>1$. For a given pair of positive integers, define its Rodriques' descendant of type $(n,k)$ as the polynomial
$$ R_{k,n,P}(x)=d^k(P^n(x))/dx^k $$
In this talk given a positive number $A<d$, we describe the root asymptotic of the sequence $R_{[An], n, P}(x)$ when $n\to\infty$. The answer is expressed through a rather explicit harmonic function related to a rational curve obtained as the result of application of the saddle-point method to the Cauchy formula for higher derivatives.