A good polynomial expansion of Bessel $I$ of $\log(x)$

Question

I am trying to find a good polynomial expansion of Modified Bessel function of Log(x); $$I_{i}(\operatorname{Log}(x))$$ for $i=(0,10)$. On plotting it on Mathematica as a function of $x$, I find the trend to be linear for $x>1$, but I was wondering if there is any good literature on these expansions with greater accuracy.

Answer 1

Probably not an answer.

For easier notation, let $x=1+t$.

You probably observed that

$$I_n(\log (t+1))=\frac{t^n}{(2 n)\text{!!}}\left(1-\frac n 2 t+\sum_{i=2}^\infty a_i^{(n)} t^i \right)$$ I have not been able to identify the patterns for the coefficients $a_i^{(n)}$ which appear in $$P_n=\sum_{i=2}^\infty a_i^{(n)} t^i $$ $$\left( \begin{array}{cc} n & P_n \\ 0 & \frac{t^2}{4}-\frac{t^3}{4}+\cdots \\ 1 & \frac{11 t^2}{24}-\frac{7 t^3}{16}+\cdots \\ 2 & t^2-t^3+\cdots \\ 3 & \frac{29 t^2}{16}-\frac{65 t^3}{32}+\cdots \\ 4 & \frac{173 t^2}{60}-\frac{73 t^3}{20}+\cdots \\ 5 & \frac{101 t^2}{24}-\frac{287 t^3}{48}+\cdots \\ 6 & \frac{81 t^2}{14}-\frac{64 t^3}{7}+\cdots \\ 7 & \frac{731 t^2}{96}-\frac{849 t^3}{64}+\cdots \\ 8 & \frac{349 t^2}{36}-\frac{665 t^3}{36}+\cdots \\ 9 & \frac{481 t^2}{40}-\frac{1991 t^3}{80}+\cdots \\ 10 & \frac{482 t^2}{33}-\frac{359 t^3}{11}+\cdots \end{array} \right)$$

I suppose that some polynomial fit could be interesting to perform since these coefficients vary in avery smooth manner.