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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!

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  • $\begingroup$ Use the integral representation 13.2.1. $\endgroup$
    – Gonçalo
    Feb 4 at 6:53
  • $\begingroup$ Thanks, @Gonçalo. That formula is for $b > a > 0$. What if $a > b > 0$? $\endgroup$ Feb 4 at 13:51
  • $\begingroup$ I don't know. ${}$ $\endgroup$
    – Gonçalo
    Feb 4 at 16:28
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    $\begingroup$ 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. $\endgroup$
    – Gonçalo
    Feb 5 at 18:09
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    $\begingroup$ Thanks very much @Gonçalo , edited! $\endgroup$ Feb 5 at 18:57

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