I don't have experience with hypergeoemtric functions, but wish to compute the following limit:


$\lim\_{x→\infty}{F([1],[a,b];-\frac{x^2}{4})}$, where $a,b$ are non-integer real parameters.


I tried to use Maple to calculate the limit and the result is 0. I tried to prove it or calculate it by hand and use an integral representation and some standard transformations but could not get the result.


Any help would be appreciated. Thanks!!



Limit of a hypergeometric function(1F2)
The large-$x$ limit is only zero if $a+b>3/2$:
$$\_1{F}\_2({1}; {a, b}; -x^2/4)=$$
$$=\sqrt{\tfrac{1}{\pi}}\Gamma (a) \Gamma (b) \sin \left(\tfrac{\pi}{2} (a+b-\tfrac{1}{2})-x\right)(2/x)^{a+b-3/2}+{\cal O}(1/x^2)$$


For example, when $a=b=1/2$ the amplitude of the oscillations increases as $\sqrt x$ (left plot), and when $a=b=3/4$ the amplitude does not decay (right plot):


![](https://ilorentz.org/beenakker/MO/1F2abonehalf.png)
![](https://ilorentz.org/beenakker/MO/1F2abthreequarters.png)


