# Approximation formula for the Kummer function ${}_1F_1(a, b, x)$ for large $x$?

The question here: Asymptotics of Kummer Hypergeometric Function in first argument addresses asymptotics for the first argument of a Kummer function. However, it is a very special case, and does not address the question of asymptotics in the last argument. So, I have been trying to get an answer to how the approximation formula for $${}_1F_1(a, b, x)$$ is derived for the case of large $$x$$. (The suggested approach in the comment does not apply for $$a>b$$.)

According to 13.1.4 of Abramowitz and Stegun, as $$x \rightarrow\infty$$,we get $${}_1F_1(a, b, x) = \frac{\Gamma(b)}{\Gamma(a)}\exp{(x)}x^{a-b}\{1 + {\mathcal O}(1/x)\},$$ but I am trying to figure out how this result arises, and in particular what would be a good approximation for $${}_1F_1(a, b, x)$$ for large $$x$$.

A similar approximation is stated in 13.2.(iv) in https://dlmf.nist.gov/13.2#iv where the result is in the form of $$\mathbf M(a,b,x) = \frac{{}_1F_1(a,b,x)}{\Gamma(b)}$$.

For numerical approximations to the $${}_1F_1(a, b, x)$$, I need to know how to derive this relation for large $$x$$. I am looking for a general $$a$$, $$b$$ and $$x>0$$.

• Thanks, @Gonçalo. That formula is for $b > a > 0$. What if $a > b > 0$? Feb 4 at 13:51
• I don't know. ${}$ Feb 4 at 16:28
• 13.2.(iv) in DLMF refers to $\mathbf{M}(a,b,z)$, defined in 13.2.4 as $M(a,b,z)=\Gamma(b)\mathbf{M}(a,b,z)$. Therefore, both approximations are equivalent. Feb 5 at 18:09