TITLE: Why are some mathematical constants irrational by their continued fraction while others aren't?
QUESTION [8 upvotes]: Catalan's Constant and quite a few other mathematical constants are known to have an infinite continued fraction (see the bottom of that webpage). On  wikipedia (I'm sorry, I can't post anymore hyperlinks because of my low rep.), a condition for irrationality of that continued fraction is given (see 'generalized continued fractions'). It is said that a given continued fraction converges to an irrational limit if $ b_n > a_n $ in the continued fraction $b_0 + a_0/(b_1 + a_1/(b_2 + a_2/(...b_n+a_n)))$ for some sufficiently large $n$. In the webpage I provided you with, however, the degree of the polynomial $a_n$ of the continued fractions is bigger than the one of $b_n$. Therefore, all values of $b_n$ will never exceed all values of $a_n$, even after some large $n$. 
My question is: Why does the degree of $a_n$ need to be smaller than the degree of $b_n$ in order for a continued fraction representation of a constant to be irrational? I think I read somewhere it had to do with something like 'Tietschzes criterion' (but I'm not sure). (bonus question: Does anyone know where a proof of this 'criterion for irrationality' can be found?) 
Thanks,
Max Muller

REPLY [2 votes]: Though infinite, the Zudilin's continued fraction for the Catalan's constant is not "simple".  So, the irrationality theorem does not apply to Zudilin result.  The distinction between "simple" and "generalized" continued fractions can be found at 
http://en.wikipedia.org/wiki/Continued_fraction
Best wishes,
Dr. Fabio M. S. Lima