[math-fun] Ratio of two Bessel function values

I recently asked, on one of these networks, about the number whose continued fraction is {1,2,3,4,...} and someone was kind enough to provide an explicit answer, which I've stupidly deleted. Can it be repeated? Incidentally, is there a more general result, that a continued fraction whose partial quotients form an arithmetic progression (or a set of APs) can be expressed in terms of Bessel functions? There are a few examples which are rational functions of e. R.