This question follows this one, where the general problem has apparently no simpler form than the integral one. I focus now on the limit case: \begin{align} \int_0^T e^{-x}\frac{nI_n(x)}{x}dx=\int_0^T e^{-x}\frac{I_{n-1}(x)-I_{n+1}(x)}{2}dx \end{align} where $I_n(x)$ is the modified Bessel function of the first kind, and $n$ is an integer.

I can get the following result from the series expansion of the modified Bessel function: \begin{align} \int_0^T e^{-x}\frac{I_{n}(x)}{2}dx&=\frac{(T/2)^{n+1}}{(n+1)!}{}_2F_2[\{n+\frac{1}{2},n+1\};\{2n+1,n+2\};-2T] \\ \int_0^T e^{-x}\frac{nI_n(x)}{x}dx&=\frac{(T/2)^{n}}{n!}{}_2F_2[\{n+\frac{1}{2},n\};\{2n+1,n+1\};-2T] \end{align}

I have some trouble to transforming this expression. I know from the answer to this post that there exist some polynomials of order $n$, $P_n(T)$ and $Q_n(T)$, such that: \begin{align} \int_0^T e^{-x}\frac{nI_n(x)}{x}dx=\Big(1+\frac{e^{-T}}{T^{n-1}}\big(P_{n-1}(T)I_0(T)+Q_{n-1}(T)I_1(T)\big)\Big) \end{align} which looks somehow to the result of an integration by parts. However I don't manage to use the derivation relation for the Bessel function, or the identities (especially number 2) with the hypergeometric function here to obtain the expression of the polynomials.

Furthermore, I have some hints (from numerical evaluations of the formula of a related model) that this could be written in terms of error functions, powers and exponentials. I have troubles to use the previous identities.

What would be an effective method to get more informations from this integral?