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I am working through a result of C.L. Siegel that uses the continued fraction:

$$ \frac{v}{v'} = \lambda + 1 + \cfrac{z}{\lambda + 2 + \cfrac{z}{\lambda + 3 + \cfrac{z}{\lambda + 4 + \cfrac{z}{\ddots}}}}$$

Where $v$ satisfies $zv'' + (\lambda + 1)v' = v$ , a linear second order differential equation.

It's developed by successive differentiation: $$zv^k + (\lambda + k - 1)v^{k - 1} = v^{k - 2} , k=2,3,...$$

So that one finally forms: $$\frac{v^{k - 1}}{v^k} = \lambda + k + \frac{z}{(\frac{v^k}{v^{k + 1}})}, k= 1,2,...$$ Hence (ultimately) some idea can be gained as to why the ratio of two Bessel Functions gives rise to the continued fraction constant, (take $\lambda$ = 0 and $ z = 1$). Siegal had a proof that this number was transcendental in 1929, but the intermediate result seems interesting.

Reference: "Transcendental Numbers" by Andrei Shidlovskii , translated by Neal Koblitz (1989)

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Not to be too pointed, but what is the question here? –  Steven Stadnicki Oct 23 '13 at 21:16
    
Why does the ratio of two Bessel Functions gives rise to the continued fraction constant, and why is it a transcendental number. –  Alan Oct 23 '13 at 22:25
    
Haven't you answered that yourself? The Bessel functions are the solutions to the differential equation you gave, and the successive-differentiation approach shows how one can build the continued fraction expression for the ratio of $\nu_0$ and $\nu_1$. –  Steven Stadnicki Oct 24 '13 at 0:46
    
You're right, although the second question -- proving the constant is transcendental is considerably more difficult. I'll move this to Wikipedia soon. –  Alan Oct 24 '13 at 2:21

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