Calculate an infinite continued fraction Is there a way to algebraically determine the closed form of any infinite continued fraction with a particular pattern? For example, how would you determine the value of $$b+\cfrac1{m+b+\cfrac1{2m+b+\cfrac1{3m+b+\cdots}}}$$?

Edit (2013-03-31):
When $m=0$, simple algebraic manipulation leads to $b+\dfrac{\sqrt{b^2+4}}{4}$. The case where $m=2$ and $b=1$ is $\dfrac{e^2+1}{e^2-1}$, and I've found out through WolframAlpha that the case where $b=0$ is an expression related to the Bessel function: $\dfrac{I_1(\tfrac2m)}{I_0(\tfrac2m)}$. I'm not sure why this happens.

Reference:  
D. H. Lehmer, "Continued fractions containing arithmetic progressions",  Scripta Mathematica vol. 29, pp. 17-24

Theorem 1:
  $$
b+\frac{1}{a+b\quad+}\quad\frac{1}{2a+b\quad+}\quad\frac{1}{3a+b\quad+}\quad\dots
= \frac{I_{b/a-1}(2/a)}{I_{b/a}(2/a)}
$$  
Theorem 2
  $$
b-\frac{1}{a+b\quad-}\quad\frac{1}{2a+b\quad-}\quad\frac{1}{3a+b\quad-}\quad\dots
= \frac{J_{b/a-1}(2/a)}{J_{b/a}(2/a)}
$$  

and other results...
