## On the generating function of the integer part: $$[n\alpha{}+ \gamma{}]$$.(English)Zbl 0778.11039

Consider the function $$G(z,w)=G_{\alpha,\gamma}(z,w)= \sum^ \infty_{n=1} z^ n w^{[n\alpha+\gamma]}$$. In case $$\gamma=0$$, this series is related to interesting continued fraction expansions and transcendental numbers and motivated work on Mahler’s method in transcendence theory. A somewhat special case of the main theorem of the paper is the identity $G(z,w)={z \over {1-z}}+ {{1-w} \over w} \sum^ \infty_{n=0} {{(-1)^{n+1} z^{t_ n^{**} w_ n^{**}}} \over {(1-z^{q_ n} w^{p_ n})(1-z^{q_{n+1}}w^{p_{n+1}})}},$ where $$p_ n/q_ n$$ are the convergents to the continued fraction expansion of $$\alpha$$ and $$t_ n^{**}$$, $$s_ n^{**}$$ arise algorithmically in a similar way from the non-homogeneous diophantine approximation problem of minimising $$| n\alpha+\gamma-m|$$. In particular, $$t_ n^{**}\alpha+\gamma-s_ n^{**}$$ are best one-sided approximants in the appropriate range and solve $$| t_ n^{**} \alpha+\gamma- s_ n^{**}|<1/t_ n^{**}$$. The non-homogeneous approximation problem is interesting and subtle in its own right and analysed in full. As in the case $$\gamma=0$$, the identity and certain continued fractions follow from a functional equation relating $$G_{\alpha,\gamma}(z,w)$$ and $$G_{\alpha^{-1},-\gamma\alpha^{- 1}}(w,z)$$. There are applications to transcendence theory. For example, if $$\alpha$$ is irrational and $$\limsup q_{n+1}/q_ n>3$$, then $$G_{\alpha,\gamma}(1/p,1/q)$$ is transcendental.
Another pretty application is a new proof of the following theorem of Fraenkel. Suppose $$\alpha>1$$, $$\alpha$$ irrational, $$0\leq\gamma<1$$ and $$n\alpha+\gamma$$ never integral. Then the sequences $$\{[n\alpha+\gamma]\} ^ \infty_{n=1}$$ and $$\{[n\alpha'+\gamma']\} ^ \infty_{n=1}$$ partition the positive integers if and only if $$1/\alpha + 1/\alpha'=1$$ and $$\gamma/\alpha + \gamma'/\alpha'=0$$.

### MSC:

 11J91 Transcendence theory of other special functions 11J20 Inhomogeneous linear forms 11J70 Continued fractions and generalizations 11B25 Arithmetic progressions
Full Text: