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 $x>0$.
Any suggestions? Thanks in advance!
