Monotihillgfur

Mh . Math. 111, 119—126 (1991) ||Ч|И111Ш11СП111

© by Springer-Verlag 1991

Printed in Amtria

Continued Fractions for Some Alternating Series

By

J . L. Davison, Sudbury and J. O. Shallit^, Waterloo (Received 17 October 1990)

Abstract . We discuss certain simple continued fractions that exhibit a type of "self-similar" stracture: their partial quotients are formed by perturbing and shifting the denominators of their convergents. We prove that all such continued fractions represent transcendental numbers. As an application, we prove that Cahen*s constant

f > 0 O, — 1

is transcendental. Here (S„) is Sylvester's sequence defined by Sq = 2 and S„+i = S^ — — S^ + 1 for я ^ 0. We also explicitly compute the continued fraction for the ber C; its partial quotients grow doubly exponentially and they are all squares.

I . Introduction

In this paper we discuss certain continued fractions that exhibit a type of "self-similar" structure: their partial quotients are formed by perturbing and shifting the denominators of their convergents. The real numbers represented by these continued fractions can also be described as the sum of an alternating series. We prove the transcendence of the numbers by an appeal to Roth's theorem. As an example, we prove that Cahen's constant is transcendental.

n . The Main Residt

If Oq is an integer and a,, öj» ••• is an infinite sequence of positive integers, then, as usual, we let

x = [ao, aj, a2, ...]

* Research supported in part by NSF grant CCR-8817400 and a Walter Burke award from Dartmouth College.