Growth of transcendental entire solution of some \(q\)-difference equation (Q1812280)

The authors consider the \(q\)-linear functional equation \(qzf(qz)+(1-Az)f(z)=1\) where \(A=e^{2\pi i\alpha}\) and \(q=e^{2\pi i\beta}\), with \(\alpha,\beta\) real. It is known that for suitable values of \(\alpha\) and \(\beta\) (with \(\beta\) irrational) this equation has an entire transcendental solution \(f\). Here it is shown that under certain conditions on \(\beta\) entire transcendental solutions \(f\) have positive order of growth. This is in contrast to the situation for \(q\)-linear functional equations with \(|q|<1\), where entire transcendental solutions \(f\) are known to satisfy \(\log M(r,f)\sim c(\log r)^2\) for some \(c>0\).