I would like to find an asymptotic expansion for the hypergeometric function

$$ _{2}F_{2}\left(a,b;c,d;z\right),\quad a,b,c,d\in\mathbb{R}. $$

The parameters are fixed. $z$ is real and $z\rightarrow +\infty$.

Could someone shed light on it?


1 Answer 1


$$_{2}F_{2}\left(a,b;c,d;z\right)=\frac{\Gamma (c) \Gamma (d)}{\Gamma (a) \Gamma (b)}e^z z^{a+b-c-d}\left(1+{\cal O}(z^{-1})\right)$$

As an example, the plot shows $_{2}F_{2}\left(a,b;c,d;z\right)$ (blue) and the asymptotics (gold) for $a=1,b=2,c=3,d=4$.

  • $\begingroup$ Thank you! Could you provide me with some reference book? $\endgroup$
    – axl
    Commented Feb 19, 2020 at 14:59
  • 1
    $\begingroup$ for example researchgate.net/publication/… $\endgroup$ Commented Feb 19, 2020 at 15:14

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