A Generating Function Technique for \lfloorn\alpha\rfloor and Related Sequences
by
Kevin O'Bryant
University of Illinois

Some generating functions of some sequences, including \sumz^{\lfloor2n/x \rfloor- \lfloorn/x \rfloor} and \sum(\lfloorn/x+1 \rfloor^-z), can be written as a sum over the rationals in the interval (0, x). Such representations are useful in analyzing when such sequences partition the positive integers (as in Beatty's Theorem), and have brought forward a periodic aspect of the generating functions involved. The Lambert series for Euler's Phi-function is a simple consequence, as are some connected expressions for other types of series. We will also show that \sigma_z(n):=\sum_d|n d z^d satisfies \piz/(4+4z²)=\sum_n=0^\infty (-1)ⁿ \sigma_z(2n+1)/(2n+1) for |z| < 1.

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Date received: April 5, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caew-66.