How is the following integral related to confluent hypergeometric functions? I am solving an integral that appears in a physics paper.
$$
-\int_0^{\infty}dt\,\frac{e^{-t}}{t}\bigg[\bigg(1+\frac{3}{N}t\bigg)^N-1\bigg]
$$
The paper does not give the full solution, it only gives the behavior of the solution when $N$ is big
$$
-\frac{3}{2}\sqrt{\frac{2\pi}{N}}\bigg(\frac{3}{e^{2/3}}\bigg)^N
$$
I have not been able to get this big $N$ behaviour but I have a better thing, I solved the integral exactly. The solution is
$$
-\int_0^{\infty}dt\,\frac{e^{-t}}{t}\bigg[\bigg(1+\frac{3}{N}t\bigg)^N-1\bigg]=-\sum_{m=1}^N\frac{N!}{(N-m)!m}\bigg(\frac{3}{N}\bigg)^m
$$
I want to obtain the alleged big $N$ behaviour. There is only one thing said in the paper about how this is done. The author says that the integral can be expressed in terms of confluent hypergeometric functions. 
I have tried to look in the wiki page for confluent hypergeometric functions but there is nothing readily useful with my level of knowledge. Thus, does anybody here have a clue how the integral above is related to confluent hypergeometric functions?
The paper in question is http://arxiv.org/pdf/hep-ph/0505034v1.pdf. Check the footnote at page 15.
 A: Let: $\mathrm{aa : =} \frac{x^m n!}{(- m + n) !m}$
Then Hypergeometric summing with respect to m you get: 
$\mathrm{hypergeom} \left( \left\{ - n + 1 \hspace{0.17em} \mathrm{,
\hspace{0.17em}} \hspace{0.17em} 1 \hspace{0.17em} \mathrm{, \hspace{0.17em}}
\hspace{0.17em} 1 \right\}, \{ 2 \}, - x \right) nx$   
For the proof we need
$Definition. Pochhammer: \left(z\right)_{k}=\frac{\Gamma\left(z+k\right)}{\Gamma\left(z\right)}$
  for $z\in\mathbb{R}$
and when $z\in\mathbb{Z}_{+}$ use:   
$\left(z\right)_{k}=\frac{\left(z+k-1\right)!}{\left(z-1\right)!}$
$\left(-z\right)_{k}\left(-x\right)^{k}=\begin{cases}
x^{k}\cdot{\frac{\left(z\right)!}{\left(z-k\right)!}} & k\leq\left(z\right)\\
0 & k>\left(z\right)
\end{cases}$
The last is a polynomial. This relation can be derived by examining the poles and zeros in the Pochhammer/Gamma definition above. This particular use of the Gamma definition to extend the Pochhammer symbol seems to be lacking in both DLMF and Wikipedia. One can use a standard table like [http://functions.wolfram.com/GammaBetaErf/Pochhammer/27/01/||Wolfram-Pochhammer] .
$Theorem. \sum_{m=1}^{N}\frac{N!}{(N-m)!m}x^{m}=N\cdot x\cdot Hypergeometric(\{\text{−}N+1,1,1\},\{2\},\text{−}x)$
Proof.
Like a lot of other operations with Hypergeometric functions this can be considered a rewrite; but rewrites can embed a problem into a form that embeds calculations into domains that are well researched.  Also notice that since it is a rewrite the fact that $x=\frac{3}{N}$ is irrelevant.
$ Hypergeom(\{\text{−}N+1,1,1\},\{2\},\text{−}x)={\displaystyle \sum_{k=0}^{\infty}}\frac{(-N+1)_{k}\left(1\right)_{k}\left(1\right)_{k}}{\left(2\right)_{k}}\frac{\left(-x\right)^{k}}{k!}$
Examining our case term-wise:
$\left(1\right)_{k}=k!$
$\left(2\right)_{k}=\left(k+1\right)!$
$\frac{\left(1\right)_{k}}{\left(2\right)_{k}}=\frac{1}{k+1}$
$\left(-N+1\right)_{k}\left(-x\right)^{k}=\begin{cases}
x^{k}\cdot{\frac{\left(N-1\right)!}{\left(N-1-k\right)!}} & k\leq\left(N-1\right)\\
0 & k>\left(N-1\right)
\end{cases}$
Thus the Hypergeometric term is:
$\frac{\left(N-1\right)!\cdot\left(k\right)!\cdot\left(k\right)!}{\left(N-1-k\right)!\cdot\left(k+1\right)!}\frac{x^{k}}{k!}=\frac{\left(N-1\right)!}{\left(N-1-k\right)!\cdot\left(k+1\right)}x^{k}$
Now we assign $m=k+1$ and sum
${\displaystyle \sum_{m=1}^{\infty}}\frac{\left(N-1\right)!}{\left(N-m\right)!\cdot\left(m\right)}x^{m-1}$
and add some indexing corrections to get
$N\cdot x\cdot Hypergeometric(\{\text{−}N+1,1,1\},\{2\},\text{−}x)$
$=N\cdot x\cdot{\displaystyle \sum_{m=1}^{\infty}}\frac{\left(N-1\right)!}{\left(N-m\right)!\cdot\left(m\right)}x^{m-1}=\sum_{m=1}^{N}\frac{N!}{(N-m)!m}x^{m}$
