Irrationality Exponents of Semi-regular Continued Fractions

Let [b0, b1, b2, ... , bn, ...] be the expansion of alpha E R N Q in regular continued fraction, and let pn/qn be the convergents of this continued fraction. It is known that the irrationality exponent of alpha is given by the formula mu (alpha) = 2 + lim supn ->infinity (log bn+1/ log qn) . We prove that this formula remains valid for semi-regular continued fractions satisfying certain conditions. In particular, it remains valid for the nearest integer and singular continued fractions. We also show that it is no longer valid for the negative and farthest integer continued fractions. In application, examples of exact computation of irrationality exponents of semi-regular continued fractions are given.