Any idea how to show that the following 2 continued fraction representations have the same limit as $n \mapsto \infty$ and $x<0<a<b$?
$$ \frac{xa}{b-x+} \frac{x(a+1)}{b+1-x+}\cdots\frac{x(a+n)}{b+n-x}$$ $$ \frac{xa}{b-x+} \frac{x(a+1)}{b+1-x+}\cdots\frac{x(a+n)}{b+n}$$
Clearly, this is a continied fraction inicialized with 0 and x respectively, where the first representation converges from below and the second one from above to some limiting point.
If I am not mistaken, the answer is given in https://en.m.wikipedia.org/wiki/Generalized_continued_fraction. The reason why this holds is that the continued fraction is convergent and hence for $n$ big enough maps any point of a complex plain to a given neighborhood of the solution, which is in this case the first serie.