A note on the function \(\sum ^{\infty}_{n=1}[nx+y]/n!\) (Q1094444)

The author considers series \(\sum^{\infty}_{n=1}(Int(nx+y))/n!\) with x,y\(\in {\mathbb{R}}\). First x is supposed to be rational. He shows the series may be written in the form \(a_ 0+\sum^{q}_{k=1}a_ k \exp (\exp (2i\pi k/q))\) with \(a_ 0\in {\mathbb{Q}}\) and the \(a_ k\) algebraic over \({\mathbb{Q}}\); thanks to the Lindemann-Weierstrass theorem he can prove the transcendence of this series provided either \(x\neq 0\) or \(y\geq 1\). In the second theorem, the author proves the \({\mathbb{Q}}\)-linear independence of 1 and the \(\sum^{\infty}_{n=1}(Int(nx+y))/n!\) provided the \(x_ j\) are strictly positive and 1 is not a linear combination of them.