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We give iterative constructions for irreducible polynomials over F_q ofdegree nt^r for all nonnegative integers r, starting from irreduciblepolynomials of degree n. The iterative constructions correspond modulofractional linear transformations to compositions with power functions x^t. TheR-transform introduced by Cohen is recovered as a particular case correspondingto x^2, hence we obtain a generalization of Cohen's R-transform (t=2) toarbitrary degrees t bigger that two. Important properties like self-reciprocityand invariance of roots under certain automorphisms are deduced from invarianceunder multiplication by appropriate roots of unity. Extending to quadraticextensions of F_q we recover and generalize a recently obtained recursiveconstruction of Panario, Reis and Wang.

The R-transform as a power map and its generalisations to higher degree