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$\begingroup$ It would be even better to get an expansion of $f_n$ at $x=1$ by the Lagrange Inversion Formula, but that leads to a residue which is not easy at all to compute, as it is for the above expansion at $x=0$. $\endgroup$

Pietro Majer
– Pietro Majer

2014-01-03 22:34:05 +00:00
Commented Jan 3, 2014 at 22:34
$\begingroup$ Thanks, this is very helpful! Although, for my application, I'm going to need an expansion at the point $x=1$. $\endgroup$

Mark Lewko
– Mark Lewko

2014-01-03 22:42:20 +00:00
Commented Jan 3, 2014 at 22:42
$\begingroup$ For an expansion at $x=1$, you can also write $f_n(x)=3+\sum_{k=1}c_{n,k}(x-1)^k$ and find the coefficients recursively, by equating the series $$\frac{1}{2}(f_n-1)(2f_n-5)^n-1=x-1$$. If you only need few terms that should be fine. $\endgroup$

Pietro Majer
– Pietro Majer

2014-01-03 22:57:16 +00:00
Commented Jan 3, 2014 at 22:57

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