<s>[INST] Asymptotic approximation for confluent hypergeometric function I have the following nasty expression that I would like to expand in powers of $\frac{1}{N}$:
\begin{align}
\frac{2^{\frac{3}{2}} 3^{\frac{1}{2}} \Biggl[ \sqrt{u} \cdot \Gamma\left(\frac{2+N}{4}\right) 
                 \cdot {}_1F_1 \left( \frac{2+N}{4},\frac{1}{2},\frac{3r^2}{2u} \right) 
                 -\sqrt{6} r \cdot  \Gamma \left( \frac{4+N}{4} \right) \cdot {}_1F_1 \left( \frac{4+N}{4},\frac{3}{2},\frac{3r^2}{2u} \right) \Biggr]
}{N \cdot  u^{\frac{1}{2}} \Biggl[ \sqrt{u} \cdot \Gamma\left(\frac{N}{4}\right) \cdot {}_1F_1 \left( \frac{N}{4},\frac{1}{2},\frac{3r^2}{2u} \right) -\sqrt{6} r \cdot  \Gamma \left( \frac{2+N}{4} \right) \cdot {}_1F_1 \left( \frac{2+N}{4},\frac{3}{2},\frac{3r^2}{2u} \right) \Biggr]}
\end{align}
where you can assume that $N \in \mathbb{N}$ (but could be analytically continued to $\mathbb{R}^+$), $u \in \mathbb{R}^+$, and $r \in \mathbb{R}^+$. Furthermore, ${}_1F_1$ is the confluent hypergeometric function sometimes written as $M(a,b,z)$.
Using a different route I have obtained a value for the limit $N \to \infty$, but I'd like to a) reproduce this result using the above expression and b) find the $O\left(\frac{1}{N}\right)$ corrections. So far I have tried numerous identities from the NIST Handbook of Mathematical Functions, but I simply seem to lack the experience to make real progress.
If anyone knows a solution or has an idea of how to proceed next, I'd greatly appreciate their help.
With best regards,
Jan
 [/INST] Perhaps not a big help:  For ${}_1F_1((2+N)/4,3/2,3r^2/2u)$ we want asymptotics of ${}_1F_1(1/2+x,3/2,a)$ as $x \to 0$.  Using the series, I get
$$
{}_1F_1\left(\frac{1}{2}+x,\frac{3}{2},a\right)=
\frac{\sqrt{\pi} \mathrm{erf} (i \sqrt{a})}{2 \sqrt{-a}} + \sum_{k = 0}^{\infty} \frac{\Bigl(\Psi \Bigl(\frac{1}{2} + k\Bigr) + \gamma + 2 \operatorname{ln} (2)\Bigr) a^{k}}{(1 + 2 k) k!}   x + \operatorname{O} \bigl(x^{2}\bigr)
$$
</s>