Symmetry and specializability in continued fractions

Abstract

We study explicit continued fraction expansions for certain series. Some of these expansions have symmetry that generalizes some remarkable examples discovered independently by Kmosek and Shallit. Furthermore, we prove the following theorem: Suppose f(x) is a polynomial with integer coefficients, and consider the sum of 1/f^n(x) as n goes from 0 to infinity, where f^n denotes the n-th iterate of f. This series has a continued fraction expansion over the polynomials. The case when the partial quotients have integer coefficients is particularly interesting, since then one can obtain simple continued fractions when one substitutes integer values for x. In this case, the continued fraction expansion is called specializable. We determine all polynomials f(x) such that the sum of 1/f^n(x) has a specializable continued fraction: this holds iff f(x) satisfies one of 14 congruences.