  We prove an explicit formula for infinitely many convergents of Hurwitzian
continued fractions that repeat several copies of the same constant and
elements of one arithmetic progression, in a quasi-periodic fashion. The proof
involves combinatorics and formal Laurent series. Using very little analysis we
can express their limits in terms of (modified) Bessel functions and Fibonacci
polynomials. The limit formula is a generalization of Lehmer's theorem that
implies the continuous fraction expansions of $e$ and $\tan(1)$, and it can
also be derived from Lehmer's work using Fibonacci polynomial identities. We
completely characterize those implementations of our limit formula for which
the parameter of each Bessel function is the half of an odd integer, allowing
them to be replaced with elementary functions.
