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Motivated by the classical Rodrigues' formula, we study the root asymptoticof the polynomial sequence $$R_{[\alpha n],n,P}(z)=\frac{d^{[\alphan]}P^n(z)}{dz^{[\alpha n]}}, n= 0,1,\dots$$ where ${P(z)}$ is a fixedunivariate polynomial, $\alpha $ is a fixed positive number smaller than deg$P$, and $[\alpha n]$ stands for the integer part of $\alpha n$.  Our description of this asymptotic is expressed in terms of an explicitharmonic function uniquely determined by the plane rational curve emerging fromthe application of the saddle point method to the integral representation ofthe latter polynomials using Cauchy's formula for higher derivatives. As aconsequence of our method, we conclude that this curve is birationallyequivalent to the zero locus of the bivariate algebraic equation satisfied bythe Cauchy transform of the asymptotic root-counting measure for the latterpolynomial sequence. We show that this harmonic function is also associatedwith an abelian differential having only purely imaginary periods and thelatter plane curve belongs to the class of Boutroux curves initially introducedby Bertola.  As an additional relevant piece of information, we derive a linear ordinarydifferential equation satisfied by $\{R_{[\alpha n],n,P}(z)\}$ as well ashigher derivatives of powers of more general functions.

Rodrigues' descendants of a polynomial and Boutroux curves