Solutions of Schr\"oder-Poincar\'e's polynomial equations $f(az)=P(f(z))$ usually do not admit a simple closed form representation in terms of known standard functions. We show that there is a one-to-one correspondence between zeros of $f$ and a set of discrete functions stable at infinity. The corresponding Vi\`ete-type infinite product expansions for zeros of $f$ are also provided. This allows us to obtain a special kind of closed-form representation for $f$ based on the Weierstrass-Hadamard expansion.
The Overdeterminedness of a Class of Functional Equations
Systems of functional equations, the generalized shift, and modelling pathological functions
Some functional equations related to the characterizations of information measures and their stability
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