Arithmetic properties of solutions of certain Poincaré-type functional equations (Q679107)

Let \(f\) be a function, holomorphic at zero, satisfying a functional equation of the form \[ x^d f(x)=P(x)f(qx)+Q(x), \] where \(P\) and \(Q\) are polynomials with rational coefficients of degree at most \(d\) and \(q\in\mathbb Z\setminus \{-1,0,1\}\). Using Padé approximation of the first kind of \(f\) at zero, it is shown that if \(f^{(v)}(0)\in \mathbb Q\) for \(0\leq v\leq d\), then \(f(r)\) is irrational for \(r\in\mathbb Q^*\), if \(f(r)\) exists and if \(f\) is not a polynomial. The proof is straightforward using a variation of a method of Mahler and does not give any measure of irrationality.