How is the following integral related to confluent hypergeometric functions?

Question

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.

Answer 1

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

$N\cdot x\cdot Hypergeometric(\{\text{−}N+1,1,1\},\{2\},\text{−}x)$ $=N\cdot x\cdot{\displaystyle \sum_{m=1}^{\infty}}\frac{\left(N-1\right)!}{\left(N-m\right)!\cdot\left(m\right)}x^{m-1}=\sum_{m=1}^{N}\frac{N!}{(N-m)!m}x^{m}$