Motivated by the classical Rodrigues' formula, we study below the root asymptotic of the polynomial sequence R-[alpha n],R-n,R-P(z) = d([alpha n]) P-n(z)/dz([alpha n]), n = 0, 1, ... where P(z) is a fixed univariate 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 explicit harmonic function uniquely determined by the plane rational curve emerging from the application of the saddle point method to the integral representation of the latter polynomials using Cauchy's formula for higher derivatives. As a consequence of our method, we conclude that this curve is birationally equivalent to the zero locus of the bivariate algebraic equation satisfied by the Cauchy transform of the asymptotic root-counting measure for the latter polynomial sequence. We show that this harmonic function is also associated with an abelian differential having only purely imaginary periods and the latter plane curve belongs to the class of Boutroux curves initially introduced in Bertola (Anal Math Phys 1: 167-211, 2011), Bertola and Mo(Adv Math 220(1): 154-218, 2009). As an additional relevant piece of information, we derive a linear ordinary differential equation satisfied by {R-[alpha n],R-n,R- P(z)} as well as higher derivatives of powers of more general functions.

The isodynamic points of a plane triangle are known to be the only pair of its centers invariant under the action of the Möbius group M on the set of triangles, Kimberling (Encyclopedia of Triangle Centers, http://faculty.evansville.edu/ck6/encyclopedia). Generalizing this classical result, we introduce below the isodynamic map associating to a univariate polynomial of degree d≥3 with at most double roots a polynomial of degree (at most) 2d−4 such that this map commutes with the action of the Möbius group M on the zero loci of the initial polynomial and its image. The roots of the image polynomial will be called the isodynamic points of the preimage polynomial. Our construction naturally extends from univariate polynomials to binary forms and further to their ratios.

We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence {(d(n)/dz(n)) (R(z) expT(z))}(n=1)(infinity). Here, R(z) is a rational function with at least two poles, all of which are distinct, and T(z) is a polynomial. This is an extension of a recent measure-theoretic refinement of Polya's Shire theorem for rational functions.

Given a complex polynomial P with zeroes z(1),..., z(d), we show that the asymptotic zero-counting measure of the iterated derivatives Q((n)), n = 1, 2,..., where Q = R/P is any irreducible rational function, converges to an explicitly constructed probability measure supported by the Voronoi diagram associated with z(1),...,z(d). This refines Polya's Shire theorem for these functions. In addition, we prove a similar result, using currents, for Voronoi diagrams associated with generic hyperplane configurations in C-m.