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

Borwein, J. M.; Borwein, P. B.

J. Number Theory 43, No. 3, 293-318 (1993).

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 \(\gamma/\alpha + \gamma'/\alpha'=0\).

Reviewer: J.H.Loxton (North Ryde)

MSC:

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

Keywords:

partition of the set of positive integers; greatest integer function; special functions; continued fraction expansions; transcendental numbers; identity; non-homogeneous diophantine approximation problem; functional equation