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Using Galois theory of functional equations, we give a new proof of the mainresult of the paper "Transcendental transcendency of certain functions ofPoincar\'e" by J.F. Ritt, on the differential transcendence of the solutions ofthe functional equation R(y(t))=y(qt), where R is a rational function withcomplex coefficients which verifies R(0)=0, R'(0)=q, where q is a complexnumber with |q|>1. We also give a partial result in the case of an algebraicfunction R.

A Galoisian proof of Ritt theorem on the differential transcendence of Poincar\'e functions